Question

Verify Rolles Theorem for the function $f\left(x\right)=x^2+2x-8,x\in [-4,2].$

Answer 1

$\text{f}\left(x\right)=x^2+2x-8$
$\Rightarrow f\left(x\right)=(x+4)(x-2)$

$f^{\prime }\left(x\right)=2x+2$
We know every polynomial function is continuous and differentiable in $R$.
In particular $f$ is continuous and differentiable in $[-4,2]$
Also $f(-4)=f\left(2\right)=0$
Thus there will exist $c\in (-4,2)$ such that $f^{\prime }\left(c\right)=0$
$\Rightarrow 2c+2=0\Rightarrow c=-1\in (-4,2)$
Hence, Rolle's theorem verified.