Question

Verify Rolle's theorum for the function $f\left(x\right)=x^3-6x^2+11x-6$ in the interval $\left[1,3\right].$

Answer 1

$f\left(x\right)=x^3-6x^2+11x-6$

$f\left(x\right)$ is cont on $\left[1,3\right]$

And differentiable on $\left(1,3\right)$

Being a polyonomial,

Now $f\left(a\right)=f\left(4\right)=1-6+11-6=0$

$f\left(b\right)=f\left(3\right)=27-54+33-6=0$

$f\left(a\right)=f\left(b\right)$

Hence rolies theorem is verified