Which of the following statements are true?
I. If $f'$ does not change sign at a critical point $c$ , then $c$ is not a local extremum.
II. Every critical point is a local maximum or a local minimum.
III. $f(x) = \lvert x \rvert$ has a local minimum at 0, though $f'(0)$ does not exist.
IV. A local maximum value can be smaller than a local minimum value of the same function.

I and III onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerIII and IV onlyCorrect answerYour answerI, III and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

I is the first derivative test's third case, and III holds because $\lvert x \rvert \ge 0 = f(0)$ near 0.
IV: $x + \frac{1}{x}$ has local maximum value $-2$ and local minimum value 2. II is false: $x^{3}$ has a critical point at 0 that is neither.

Worth noting

'Local' means compared with nearby points only, which is why IV can happen.

Concept: What local maxima and minima mean
Typical timeTypical: 150 s
Your timeYours: —

The rule this testsLocal extrema and the first derivative test

Practise this chapter →

Ten questions from Maxima & Minima, at your own pace.

Explanation