Question

Verify lagrange's mean value theorem for the function
$f\left(x\right)=x^2+2x+3$ where $x\in [4,6]$

Answer 1

Consider the following question.

$f\left(x\right)=x^2+2x+3c\in (4,6)$

So f(x) being a polynomial is continuous and differentiable on $(4,6)$

So there must exist at least one real number $c\in (4,6)$ such that

$f\prime \left(c\right)=\frac{f\left(6\right)-f\left(4\right)}{6-4}$

$Step2:$

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

$f\left(6\right)=6^2+2\left(6\right)+3=51$

$f\left(4\right)=4^2+2\left(4\right)+3=27$

$f\prime \left(x\right)=2x+2$

$\therefore f^{\prime }\left(c\right)=2c+2$

$2c+2=\frac{51-27}{2}$

$2c+2=12$

$2c=10$

$c=5$

$c\in (4,6)$

Hence verified.