Practice question · Select all that apply
Let c be a critical point of f, so f'(c) = 0. Select every correct conclusion.
Hints
- Two options say opposite things about the case f'' = 0; only one can be right.
- Test the case f'' = 0 on x to the fourth at 0, which is a genuine minimum.
Show the answer
- A. If f''(c) = 0 the second-derivative test gives no verdict
- B. f''(c) > 0 means the curve is concave up at c
- C. If f''(c) > 0 then c is a local minimum
- D. If f''(c) < 0 then c is a local maximum
Why
Concave up at a flat point means the bottom of a cup, hence a minimum, and concave down gives a maximum. When f'' = 0 the test is silent: x to the fourth has a minimum there, x cubed has neither, and minus x to the fourth has a maximum, all three with f'' = 0.
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