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6-15.

Sketch a continuous function with the following properties: $f^\prime(–2) = 0$, $f^\prime(3)$ does not exist, $f^{\prime\prime}(x) < 0$ for $x < 3$, and $f^{\prime\prime}(x) > 0$ for $x > 3$. Homework Help ✎

$f^\prime(−2) = 0$

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.)

$f^\prime(3) = \text{DNE}$

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

$f^{\prime\prime}(x) < 0$ for $f^{\prime\prime}(x) < 0$ and $f^{\prime\prime}(x) > 0$ for $x > 3$

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