### Home > CALC > Chapter 5 > Lesson 5.3.3 > Problem 5-132

For each part below, what can you conclude (if anything) about

*f*(*x*) if you know the given information below? (Note: Each part is separate.) Homework Help ✎*f*′(−2) = 0 and*f*″ (−2) > 0.*f*′(*x*) < 0 when*x*> 3,*f*′(*x*) > 0 when*x*< 3, and*f*′(3) = 0.*f*″(3) > 0*f*″(3) < 0*f*″(3) = 0*f*(*x*) is continuous at*x*= 3, but not differentiable there.*f*(*x*) is defined and continuous everywhere, and has just one critical point at*x*= 2, which is a local maximum.*f*″(3) does not exist and*f*″(*x*) < 0 for*x*< 3 and*f*″(*x*) > 0 for*x*> 3.

This is a justification about whether *f*(*x*) has a local max or a local min at *x* = −2. But which one?

2nd Derivative Test.

This is a justification about whether *f*(*x*) has a local max or a local min at *x* = 3. But which one?

1st Derivative Test.

Complete the sentence: At *x* = 3, the 2nd derivative is positive so *f*(3) is ____________________________

Two words.

Refer to hint in part (c).

There might be a point of inflection on *f*(*x*) at *x* = 3. But we do not know for sure. There are two ways to find out:

1. You could check the 2nd-derivative to the left of *x* = 3. If it changes signs then f(3) is a POI, if not then it's not.

2. You could evaluate the 3rd-derivative at *x* = 3. If it is NOT zero, then f(3) is a POI. If is is zero, then this method is inconclusive.

Since we are not given any extra information, we can only say that f(3) is a CANDIDATE for a point of inflection.

Examples of points of NON-differentiability include: cusps, endpoints, jumps, holes and vertical tangents.

Which of the above attributes still possess continuity?

Condition 1: Either *f* '(2) = 0 or *f* '(2) does not exist.

Condition 2: Refer to hints in part (a) or part (b).

Refer to hint in part (e).