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 ____________________________
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).