Question

Verify Rolle's theorem for the following function: $f\left(x\right)=x^2-4x+10$ on $[0,4]$.

Answer 1

$f\left(x\right)=x^2-4x+10\forall x\in [0,4]$

Now, Rolle's theorem states that there exists a $c\in [0,4]$ such that $f^{\prime }\left(c\right)=0$ if $f\left(4\right)=f\left(0\right)$

Now, $f\left(4\right)=10$ and $f\left(0\right)=10$. Hence, first condition is satisfied.

Also, on solving the equation,

$2c-4=0$

$\therefore 2c=4$

$\therefore c=2$

Now, as $c\in [0,4]$, Rolle's theorem is satisfied.