Which of the following statements are true?
I. If $f'(x) > 0$ at every point of $(a, b)$ , then $f$ is strictly increasing on $(a, b)$ .
II. If $f$ is strictly increasing on $(a, b)$ , then $f'(x) > 0$ at every point of $(a, b)$ .
III. If $f'(x) = 0$ at every point of $(a, b)$ , then $f$ is constant on $(a, b)$ .
IV. If $f'(c) < 0$ at one point $c$ , then $f$ is decreasing on $\mathbb{R}$ .

I, II and III onlyCorrect answerYour answerI and III onlyCorrect answerYour answerI, III and IV onlyCorrect answerYour answerI and II onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

I and III are the derivative test: the sign of $f'$ throughout an interval settles the behaviour there.
II fails for $x^3$ on $(-1, 1)$ , where $f'(0) = 0$ ; IV fails because one point says nothing about the whole line.

Concept: What the derivative test says
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe derivative test

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Ten questions from Increasing & Decreasing Functions, at your own pace.

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