Verify the conditions of the Lagrange's mean value theorem for the following function in each case find a point ‘$c$’ in the interval as stated by the theorem.

$x^2-1\text{ on }[2,3]$

$f\left(x\right)=x^2-1\text{ on }[2,3]$

Since $f\left(x\right)$ is polynomial

$\therefore$ it is continuous on $[2, 3]$ and differentiable on $(2, 3)$

$f^{\mathrm{\prime }}\left(x\right)=2x\Rightarrow f^{\mathrm{\prime }}\left(c\right)=2c$

From Lagrange's mean value theorem $\mathrm{\exists }c\in (2,3)$

Such that $f^{\mathrm{\prime }}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$\Rightarrow 2c=\frac{8-3}{3-2}\Rightarrow 2c=5\Rightarrow c=\frac{5}{2}\in (2,3)$

$\therefore$ Lagrange’s theorem verified