CPM Homework Help : CALC Problem 6-15

Sketch a continuous function with the following properties: $f'(−2) = 0$ , $f'(3)$ does not exist, $f''(x) < 0$ for $x < 3$ and $f''(x) > 0$ for $x > 3$ .

Clue 1:

$f'(−2) = 0$

Translation: the slope of $f(x)$ is zero at $x = −2$ . Note: $x = −2$ is a CANDIDATE for local max or min. (It might also be a point of inflection.)

Clue 2:

$f'(3) =\text{DNE}$

Reasons why a derivative might not exist at a point:
cusp
endpoint
jump
hole
vertical tangent

Clue 3:

$f''(x) < 0$ for $x < 3$ and $f''(x) > 0$ for $x > 3$

Translation: There is a change in concavity at $x = 3$ .