Question

Verify Rolle's theorem for the following function $f\left(x\right)=x^2-5x+9,x\in [1,4]$

Answer 1

$f\left(x\right)=x^2-5x+9$
$={[x-\frac{5}{2}]}^2+9-\frac{25}{4}$
$={[x-\frac{5}{2}]}^2+\frac{11}{4}$
$f\left(1\right)=1^2-5\times 1+9=5$
$f\left(4\right)=4^2-5\times 4+9=5$
$f\left(1\right)=f\left(4\right)f\left(x\right)$ is continuous as well as differentiate in $(1,4)$
There exists $C\epsilon (1,4)$
$f^{\prime }\left(x\right)=0$
$f^{\prime }\left(x\right)=2x-5$
$x=\frac{5}{2}$