Discuss the validity of Rolle's Theorem 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.

$f$ is polynomial, so we can apply the theorem of Rolle. We have $f'(x)=8x-20$, hence  $f'(\frac{5}{2})=0$.
