Question

Verify the Lagrange's theorem for the following functions:

$f\left(x\right)=x^3$ in $[-1,1]$

Answer 1

$f\left(x\right)=x^3,f^{\prime }\left(x\right)=3x^2$ and $f^{\prime }\left(c\right)=3c^2$
Let $a=-1,b=1,f\left(a\right)=f(-1)={(-1)}^3=-1$
$f\left(b\right)=f\left(1\right)={\left(1\right)}^3.=+1.$

By Lagrange's mean value theorem, we have
$f\left(b\right)=f\left(a\right)+(b-a)f^{\prime }\left(c\right)$
$f\left(1\right)=f(-1)+(1+1)f^{\prime }\left(c\right)$
$1=-1+2f^{\prime }\left(c\right)$
$\Rightarrow 1=-1+2\left(3c^2\right)$
$\Rightarrow 1=-1+6c^{2}2=6c^2$
$2=6c^2\Rightarrow c^2=\frac{1}{3},c=\frac{1}{\sqrt{3}}ϵ[-1,1]$
Hence, Lagrange's mean theorem is verified.