Unit 5 Beta

Analytical Applications of Differentiation: the one-page sheet

5.1 The Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b) , some c inside has f′(c) equal to the average rate of change: a tangent line parallel to the secant line. Check the hypotheses every time (" f is differentiable, so it is continuous") and name the theorem.

Mean Value Theorem: if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then $f'(c)$ $= \frac{f(b) - f(a)}{b - a}$ for at least one $c$ in $(a, b)$ .
In words: somewhere inside, the instantaneous rate of change equals the average rate of change.
" $f$ is differentiable" covers both hypotheses, because differentiable implies continuous. Say it.
The theorem says a $c$ exists. It does not say where, and there may be more than one.
A corner at an endpoint is fine: differentiability is needed only on the open interval.

the graph is one unbroken, smooth curve from (a, f(a)) to (b, f(b)) , with a tangent line at each inside point the last place it touches the curve is a point where the tangent line is parallel to the secant line some c in (a, b) has f′(c) = (f(b) − f(a))/(b − a) the theorem promises nothing, and the conclusion may be false

Mean Value Theorem: If <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous on <math class="calc-m"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a, b]</annotation></semantics></math> and differentiable on <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, b)</annotation></semantics></math>, then there is at least one <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> in <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, b)</annotation></semantics></math> with <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(c)</annotation></semantics></math><wbr><math class="calc-m"><semantics><mrow><mo form="infix">=</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">= \frac{f(b) - f(a)}{b - a}</annotation></semantics></math>: the tangent line at <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> is parallel to the secant line.

5.2 Extreme Value Theorem, Global and Local Extrema, Critical Points

A function continuous on a closed interval reaches an absolute maximum and minimum there (Extreme Value Theorem). Look for them at the endpoints and at critical points, interior points where f′ is 0 or does not exist. A critical point is a candidate, not a guarantee of an extremum.

Extreme Value Theorem: if $f$ is continuous on $[a, b]$ , then $f$ has at least one absolute maximum value and at least one absolute minimum value on $[a, b]$ .
Absolute (global) maximum: $f(c) \ge f(x)$ for every $x$ in the domain or interval. Relative (local) maximum: $f(c) \ge f(x)$ for every $x$ near $c$ , on an open interval around $c$ . Minimums the same way.
Critical point: an interior point $c$ of the domain where $f'(c) = 0$ or $f'(c)$ does not exist.
Relative extrema happen only at critical points. Absolute extrema on $[a, b]$ happen at critical points or at endpoints.
If a hypothesis fails (a jump, an open interval), the theorem promises nothing; extrema may or may not exist.

the Extreme Value Theorem guarantees an absolute maximum value and an absolute minimum value on [a, b] it is also a relative extremum, where the tangent line is flat or does not exist those interior points, the critical points, plus the endpoints, are the only places to look some critical points ( x³ at 0, a corner on a rising graph) are not extrema at all

Extreme Value Theorem: If <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous on a closed interval <math class="calc-m"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a, b]</annotation></semantics></math>, then <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has at least one absolute maximum value and at least one absolute minimum value on <math class="calc-m"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a, b]</annotation></semantics></math>.
critical point: An interior point <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> of the domain of <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> where <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(c) = 0</annotation></semantics></math> or <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(c)</annotation></semantics></math> does not exist. Relative extrema can occur only at critical points.
absolute maximum: An absolute (global) maximum is a value <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c)</annotation></semantics></math> with <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c) \ge f(x)</annotation></semantics></math> for every <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in the domain or interval considered; an absolute minimum has <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c) \le f(x)</annotation></semantics></math> for every such <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.
relative maximum: A relative (local) maximum is a value <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c)</annotation></semantics></math> with <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c) \ge f(x)</annotation></semantics></math> for every <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in some open interval around <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>; a relative minimum has <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(c) \le f(x)</annotation></semantics></math> there. It is a peak or valley compared with nearby points only.

5.3 Where a Function Increases or Decreases

The sign of the derivative gives the direction of a function: f′ > 0 on an interval means f is increasing there, f′ < 0 means decreasing. Split the domain at the critical points, test each piece, and justify with the sign of f′ by name. On a graph of f′ , read the sign, not the slope.

Increasing on an interval: $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ in the interval. If $f'(x) > 0$ on the interval, $f$ is increasing there; if $f'(x) < 0$ , decreasing.
Method: find $f'$ , find where it is 0 or undefined, test the sign of $f'$ on each interval between those points.
On a graph of $f'$ : above the $x$ -axis means $f$ increasing; below means $f$ decreasing.
Justify with the sign of $f'$ by name: " $f'(x) > 0$ for $-2 < x < 1$ , so $f$ is increasing there."

each tangent line slopes up, so f is increasing on that interval f is decreasing on that interval the critical points split the line into intervals, and one test value per interval gives the sign read the sign (above or below the axis), not whether the graph of f′ goes up or down

increasing on an interval: <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is increasing on an interval if <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo><</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1) < f(x_2)</annotation></semantics></math> whenever <math class="calc-m"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo><</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1 < x_2</annotation></semantics></math> in the interval, and decreasing if <math class="calc-m"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>></mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1) > f(x_2)</annotation></semantics></math>. If <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(x) > 0</annotation></semantics></math> on an interval, <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is increasing there; if <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(x) < 0</annotation></semantics></math>, it is decreasing.

5.4 The First Derivative Test

At a critical point, look at the sign of f′ on each side. + to − : relative maximum. − to + : relative minimum. No change: no extremum. The test works where f′ is undefined too. Justify in words that name f′ and the direction of its sign change; a sign chart alone does not earn the point.

First Derivative Test: at a critical point $c$ where $f$ is continuous, if $f'$ changes from $+$ to $-$ , $f(c)$ is a relative maximum; from $-$ to $+$ , a relative minimum; no change, neither.
It works where $f'$ does not exist (corners, cusps) as well as where $f' = 0$ .
A sign chart of $f'$ organizes the work, but it is not a justification by itself: write the sentence.
On a graph of $f'$ : a relative maximum of $f$ is where the graph of $f'$ crosses the axis going down; a minimum, crossing going up. Touching without crossing gives nothing.

f rises up to c and falls after it, so f(c) is a relative maximum f falls then rises, so f(c) is a relative minimum f keeps rising (or falling) through c : no relative extremum, even if f′(c) = 0 write the sentence: " f′ changes from negative to positive at x = c , so f has a relative minimum there"

First Derivative Test: At a critical point <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> where <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous: if <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> changes from positive to negative at <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has a relative maximum at <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>; from negative to positive, a relative minimum; if <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> does not change sign, no relative extremum. A sign chart of <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> organizes the test; the justification is a sentence naming <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> and its sign change.

5.5 The Candidates Test

On a closed interval, a continuous function has absolute extrema, and they sit at endpoints or critical points. The Candidates Test evaluates f at each of those and compares. A sign change of f′ shows only a relative extremum; an absolute one needs the comparison.

Candidates Test (closed interval method): for $f$ continuous on $[a, b]$ , list the endpoints and the critical points in $(a, b)$ , compute $f$ at each, and compare.
No derivative signs needed: just values. Show the table of candidates and values in your answer.
The absolute extrema can be at endpoints. A relative maximum can lose to an endpoint.
On a calculator problem, find critical points with the graph or solver, store them, and evaluate the model at each candidate.

by the Extreme Value Theorem, an absolute maximum and minimum exist if inside, it is at a critical point, where f′ is 0 or undefined evaluate f at each one and compare: the largest value is the absolute maximum, the smallest the absolute minimum that proves a relative extremum, a local fact, not an absolute one

Candidates Test: A way to find the absolute extrema of a function continuous on <math class="calc-m"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a, b]</annotation></semantics></math>: evaluate <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> at the endpoints <math class="calc-m"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> and at every critical point in <math class="calc-m"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, b)</annotation></semantics></math>; the largest value is the absolute maximum and the smallest the absolute minimum. Also called the closed interval method.

5.6 Concavity and Points of Inflection

The sign of f″ gives the concavity: f″ > 0 (equivalently f′ increasing) means concave up, f″ < 0 means concave down. A point of inflection is where the concavity changes, so f″ must change sign; f″(c) = 0 alone is not enough. On a graph of f′ , look at where f′ turns.

Concave up on an interval: $f'$ is increasing there ( $f'' > 0$ ). Concave down: $f'$ is decreasing ( $f'' < 0$ ).
Point of inflection: a point on the graph where the concavity changes, that is, where $f''$ changes sign, or where $f'$ changes from increasing to decreasing or back.
On a graph of $f'$ : inflection points of $f$ are at the high and low points of $f'$ , not at its zeros.
Candidates are where $f'' = 0$ or $f''$ does not exist; then check the sign of $f''$ on each side.

f′ is increasing there, the slopes grow, and the graph of f is concave up f′ is decreasing, the slopes shrink, and the graph is concave down the concavity changes, so (c, f(c)) is a point of inflection no change of concavity, so no point of inflection

point of inflection: A point on the graph of a continuous function where the concavity changes, from concave up to concave down or the reverse. For a twice-differentiable <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">″</mo></mrow></mrow><annotation encoding="application/x-tex">f''</annotation></semantics></math> changes sign there; equivalently, <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> changes from increasing to decreasing or back.

5.7 The Second Derivative Test

At a critical point where f′(c) = 0 , the sign of f″(c) decides: positive gives a relative minimum, negative a relative maximum, zero gives no conclusion. State both conditions in a justification. With exactly one critical point on an interval, a relative extremum there is absolute.

Second Derivative Test: if $f'(c) = 0$ and $f''(c) > 0$ , $f$ has a relative minimum at $c$ ; if $f'(c) = 0$ and $f''(c) < 0$ , a relative maximum.
It says nothing when $f''(c) = 0$ or when $f'(c)$ does not exist. Then use the First Derivative Test.
Justify with both facts: " $f'(1) = 0$ and $f''(1) > 0$ ."
One critical point on an interval, and it is a relative minimum (maximum)? Then it is the absolute minimum (maximum) there, even on an open interval.

the tangent line at c is flat and the graph bends up away from it, so f(c) is a relative minimum the graph bends down from a flat point, so f(c) is a relative maximum the test is inconclusive; go back to the First Derivative Test that relative extremum is also the absolute extremum on the interval

Second Derivative Test: If <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(c) = 0</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">″</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f''(c) > 0</annotation></semantics></math>, then <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has a relative minimum at <math class="calc-m"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>; if <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(c) = 0</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">″</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f''(c) < 0</annotation></semantics></math>, a relative maximum. If <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">″</mo></mrow><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f''(c) = 0</annotation></semantics></math>, the test gives no conclusion.

5.8 Sketching f, f' and f''

Sketch a derivative feature by feature: where f rises, f′ is positive; where f is flat at a peak or valley, f′ crosses zero; where f is concave up, f′ rises; where f inflects, f′ turns; where f has a corner, f′ jumps. Read the same links backward to sketch f from f′ .

Read a graph of $f$ for features, not values: where it rises, falls, peaks, bends. Each feature becomes a feature of the graph of the derivative.
$f$ (sign of slope) $\to$ $f'$ (sign of value). $f$ (concavity) $\to$ $f'$ (rising or falling). Inflection of $f$ $\to$ peak or valley of $f'$ .
Going the other way, from $f'$ to $f$ : zeros of $f'$ where its sign changes are extrema of $f$ ; turns of $f'$ are inflection points of $f$ ; a constant $f'$ means $f$ is a straight line.
The same pattern links $f'$ and $f''$ : the slope of the graph of $f'$ is $f''$ .

the graph of f′ is above (below) the x -axis there the graph of f′ crosses the x -axis there the graph of f′ is rising (falling) the graph of f′ has a peak or valley there the graph of f′ jumps, with no value at that point

graph of the derivative: The graph of <math class="calc-m"><semantics><mrow><mi>y</mi><mo>=</mo><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y = f'(x)</annotation></semantics></math>, whose height at each <math class="calc-m"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is the slope of <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> there. It is above the axis where <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> rises, crosses zero at the flat peaks and valleys of <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, rises where <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is concave up, and turns at the inflection points of <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.

5.9 Connecting f, f' and f''

The three graphs carry one story. f′ gives the direction of f , and its sign changes give extrema. f″ (or whether f′ rises or falls) gives the concavity of f , and its sign changes give inflection points. Read from whichever representation you have, and name the function in every justification.

Graphical analysis: given any one of $f$ , $f'$ , $f''$ (graph, table or formula), read the features of the others.
From a graph of $f'$ : sign of $f'$ (above or below the axis) gives increasing or decreasing; direction of $f'$ (rising or falling) gives concavity.
From a table: $f'(a) = 0$ and the sign of $f''(a)$ give the Second Derivative Test; a continuous $f''$ that changes from positive to negative between two inputs gives a point of inflection between them.
Words: "increasing at a decreasing rate" means $f' > 0$ and $f'' < 0$ .

tells whether f is increasing or decreasing; a sign change of f′ is a relative extremum of f tells the concavity of f ; a sign change of f″ is a point of inflection of f give the tangent line, and the sign of f″ near a says whether it overestimates or underestimates names the function whose sign or direction you read: " f′ changes from positive to negative", " f′ is decreasing"

graphical analysis: Reading the behavior of a function from any representation of <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> or <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">″</mo></mrow></mrow><annotation encoding="application/x-tex">f''</annotation></semantics></math>: the sign of <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> gives where <math class="calc-m"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> increases or decreases, the sign of <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">″</mo></mrow></mrow><annotation encoding="application/x-tex">f''</annotation></semantics></math> (or whether <math class="calc-m"><semantics><mrow><mrow><mi>f</mi><mo lspace="0" rspace="0">′</mo></mrow></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> rises or falls) gives concavity, and sign changes locate relative extrema and points of inflection.

5.10 Introduction to Optimization

Optimization finds the absolute maximum or minimum of a quantity in context. Write the objective function, reduce it to one variable using the given facts, find its domain, find the critical points, and justify the absolute extremum with candidates or one-critical-point reasoning. Answer in context, with units.

Optimization: finding the absolute maximum or minimum of a quantity in context.
The objective function is the quantity to optimize, written in one variable with a domain.
Justify the absolute extremum: Candidates Test on a closed interval, or one critical point that is a relative extremum.
Answer the question asked: the dimensions or the price, with units, not just $x$ .

name that quantity and write the objective function for it use the given fact (a fixed perimeter, a fixed product) to write one variable in terms of the other find the domain: lengths and amounts are not negative the absolute extremum, justified, answered in context with units

optimization: Finding the absolute maximum or minimum value of a quantity in context. The quantity is written as an objective function of one variable on a sensible domain, and its absolute extremum is found and justified with derivatives.

5.11 Solving Optimization Problems

Solve an optimization problem in a fixed order: draw and label, write the objective function, use the constraint to reduce it to one variable, find the domain, find the critical points, justify the absolute extremum with candidates or one-critical-point reasoning, and answer in context with units.

Constraint: a fixed condition (a perimeter, a volume, a budget) that links the variables. Use it to write the objective in one variable.
Steps: draw, label, objective, constraint, one variable, domain, critical points, justify, answer in context.
Reject critical points outside the domain (the cut $x = \frac{20}{3}$ inches on a 10-inch side).
Calculator problems: solve $C'(x) = 0$ with the solver or a graph, store the value, and compare candidates.

you can see which quantities are fixed and which change the constraint lets you eliminate a variable: the objective becomes a function of one variable you know the endpoints and which critical points count the absolute extremum, justified, stated in context with units

constraint: In an optimization problem, a fixed condition that links the variables, such as a given perimeter, volume or budget. It is used to write the objective function in terms of one variable.

5.12 Behavior of Implicit Relations

For an implicit curve, dy/dx = N/D . Horizontal tangents: N = 0 , D ≠ 0 , on the curve. Vertical tangents: D = 0 , N ≠ 0 , on the curve. Both zero: no conclusion. At a horizontal tangent, the sign of d²y/dx² tells whether the curve has a local high or low point.

Horizontal tangent: numerator of $\frac{dy}{dx}$ is 0, denominator is not, point is on the curve. Vertical tangent: denominator 0, numerator not 0, point on the curve.
To find the points: set the numerator (or denominator) to 0, solve for one variable, substitute into the curve's equation, then check the other part is not 0.
$\frac{d^2y}{dx^2}$ : differentiate $\frac{dy}{dx}$ (quotient rule), then substitute the point and the value of $\frac{dy}{dx}$ there.
Write " $= 0$ " after differentiating, and apply $\frac{d}{dx}$ to both sides.

dy/dx = (N(x, y))/(D(x, y)) , a slope at each point (x, y) of the curve the tangent line is horizontal there the tangent line is vertical there the formula decides nothing at that point the curve has a local high point ( < 0 ) or low point ( > 0 )

horizontal tangent: A point of a curve where the tangent line is horizontal: <math class="calc-m"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{dy}{dx} = 0</annotation></semantics></math>. For an implicit curve with <math class="calc-m"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mi>N</mi><mi>D</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{dy}{dx} = \frac{N}{D}</annotation></semantics></math>, the numerator <math class="calc-m"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> is 0, the denominator <math class="calc-m"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is not, and the point lies on the curve. (Where <math class="calc-m"><semantics><mrow><mi>D</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">D = 0</annotation></semantics></math> and <math class="calc-m"><semantics><mrow><mi>N</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">N \ne 0</annotation></semantics></math>, the tangent line is vertical instead.)