Question

Verify Mean Value Theorem,if $f(x)=x^2-4x-3$ in the interval $[a,b]$,where $a=1$ and $b=4$

Answer 1

The given function is $f(x)=x^2-4x-3$
$f$, being a polynomial function, is continuous in $[1,4]$ and is differentiable in $(1,4)$ whose derivative is $2x−4$.
$f(1)=1^2-4\times1-3=-6$
$f(4)=(4)^2-4\times4-3=-3$
$∴\frac{f(b)-f(a)}{b-a}=\frac{f(4)-f(1)}{4-1}$
$=\frac{-3-(-6)}{3}=\frac{3}{3}=1$
Mean Value Theorem states that there is a point $c∈(1,4)$ such that $f'(c)=1$
$f'(c)=1$
$⇒2c-4=1$
$⇒c=\frac{5}{2}$,where $c=\frac{5}{2}∈(1,4)$
Hence, Mean Value Theorem is verified for the given function.