### Home > CALC > Chapter 6 > Lesson 6.1.1 > Problem 6-15

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. Homework Help ✎

*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.)

*f* '(3) = DNE

Reasons why a derivative might not exist at a point:

cusp

endpoint

jump

hole

vertical tangent

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

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