Behaviour of $f'(x)$ and $f''(x)$

Question

If $f'(x)$ is decreasing, is it always true that $f''(x)$ is also decreasing?

I want to make sure I understand this concept thoroughly.

Thanks.

score 2 · Answer 1 · answered Nov 26 '15 at 09:58

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Forget $f$ for a second and ask a different question: If a function $g$ is decreasing, does that mean that $g'$ is decreasing as well? Now replace $g$ with $f'$, and you have your answer.

answered Nov 26 '15 at 09:58

Arthur

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Thanks for clarifying Arthur, but my question was more so directed towards the second derivative. If the first derivative of a function is decreasing, is the second derivative also decreasing? – user292974 Nov 26 '15 at 10:04
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@user292974 Every differentiable function is continuous, so it has an antiderivative. So Arthur's hint is what you need. – egreg Nov 26 '15 at 10:06

score 1 · Answer 2 · answered Nov 26 '15 at 10:15

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Plot $f(x)=\sin(x)$. Find $f'(x),f''(x)$.

What can you infer about the question analysing $f''(x),f'(x)$ between $\left[\frac\pi2,\pi\right]$?

Geogebra easily calculates derivatives (just input $f'(x)$ or $f''(x)$ after defining $f$), pick random functions, experiment for yourself and see.

answered Nov 26 '15 at 10:15

Jesse P Francis

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Jesse, that definitely helped. Thank you. – user292974 Nov 26 '15 at 10:36
@user292974, as you mentioned in your question, to understand the concept properly, you should try understand Arthur's answer:) – Jesse P Francis Nov 26 '15 at 10:38

score 0 · Answer 3 · answered Nov 26 '15 at 10:41

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f(x)=$-x^4/12$ is a counter example too .

answered Nov 26 '15 at 10:41

Sota Antonino

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Behaviour of $f'(x)$ and $f''(x)$

3 Answers3