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Guys this is the question and I tried to solve its first part but I am not sure if it is correct or not and also I am unable to solve the second part, kindly help me.

Question: Part (1): Discuss the validity of Rolle's Theorem for the function $f(x)=4x^2-20x+29$ over the interval $[1,4]$
Part (2): Find $c$, if possible.

My Attempt for Part (1): I calculated the $f(1)$ and $f(4)$, both were equal to $13$, hence in my view the Rolle's Theorem is valid for this function

Kindly correct me if I am wrong and also kindly tell me how can the Part (2) of this question be solved. Thanks in advanced.

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  • $\begingroup$ You are correct with Part (1) $\endgroup$ – Peter Jan 20 '17 at 20:16
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I agree that $f(1)=f(4)=13$, but you should also mention that $f$ is continuous on $[1,4]$ and differentiable on $(1,4)$.

For the second part, the question is presumably asking you to find a point $c\in (1,4)$ such that $f^{\prime}(c)=0$. So compute $f^{\prime}(x)$, set it equal to zero, and solve for $x$.

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  • $\begingroup$ Thanks, going to do it, thanks a lot $\endgroup$ – Khubaib Khawar Jan 20 '17 at 20:29
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$f$ is polynomial, so we can apply the theorem of Rolle. We have $f'(x)=8x-20$, hence $f'(\frac{5}{2})=0$.

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  • $\begingroup$ Hey, may I know that from where did $5/2$ come? $\endgroup$ – Khubaib Khawar Jan 20 '17 at 20:29

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