A table of f and its derivative: the…

Name: ______________________ Date: ____________ A table of f and its derivative: the theorems and a tangent line (9 points)

Table

The function

f

is twice differentiable for all real numbers. The table gives values of

f

and its derivative

f'

at selected values of

x

Values of $f(x)$ and $f'(x)$ at selected $x$
$x$	0	1	3	4	6
$f(x)$	2	5	5	2	3
$f'(x)$	4	1	-2	-3	2

(a) Must there be a value $c$ , for $1 < c < 3$ , such that $f'(c) = 0$ ? Justify your answer. [2 points]
(b) Must there be a value $c$ , for $0 < c < 6$ , such that $f(c) = 4$ ? Justify your answer. [2 points]
(c) Write an equation for the line tangent to the graph of $f$ at $x = 4$ , and use it to approximate $f(4.2)$ . [2 points]
(d) Must there be a value $c$ , for $4 < c < 6$ , such that $f''(c) = \frac{5}{2}$ ? Justify your answer. [3 points]

The question