Question

Verify Lagrange's mean value theorem for the following function on the indicated interval. In each case find a point ${}^{\prime }{c}^{\prime }$ in the indicated interval as stated by the Lagrange's mean value theorem:
$f\left(x\right)=x^2-3x+2\text{on}[-1,2]$

Answer 1

$f\left(x\right)$ is polynomial function and is continuous at $[-1,2]$ and differential at $(-1,2)$.

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

$f^{\prime }\left(x\right)=2x-3$

According to Lagrange's Mean Value theorem;

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

$2c-3=\frac{(4-6+2)-(1+3+2)}{2+1}$

$2c-3=\frac{0-6}{3}$

$2c-3=-2$

$2c=1$

$c=\frac{1}{2}\in [-1,2]$