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Can any anyone clarify or prove that if the derivative of a function $f$ is strictly positive then the function $f$ is strictly monotone increasing. I am really sure that the converse is not true as the converse will not be true for the function $f(x)=x^3$. I thank every one prove the first part of the this problem for me.

Amzoti
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LoveMath
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1 Answers1

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Take a look at the mean value theorem. If the function $f$ is not strictly increasing, what does this theorem tell you?

Alistair Savage
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  • Do you mean that by mean value theorem that there is $c$ such that $f^'(c)=\frac{f(x_2)-f(x_1)}{(x_2)-(x_1)}>0$? – LoveMath May 06 '13 at 01:00
  • Yes, that's right. If $f$ is not strictly increasing, you can find points $x_1$ and $x_2$ with what properties? And then was does that imply about $f'(c)$? – Alistair Savage May 06 '13 at 01:02