Let $f$ be a continuously differentiable function which satisfy $f(0)=0,f(1)=3,f(2)=7$. I want to prove that there is a real number $c$ such that $f'(c)=\pi$. Applying the mean value theorem, the derivative is $3$ somewhere in $[0,1]$ and $4$ somewhere in $[1,2]$. Now I think that since the derivative is continuous it must by the intermediate value theorem attain all values between $3$ and $4$ and therefore especially $\pi$?
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4short answer: yes – cats Dec 13 '16 at 22:36
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1Yep. I had to look up "continuously differentiable" but as it means derivatibe is continuous then, yep, your answer is sound. – fleablood Dec 13 '16 at 22:58