Question

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

Answer 1

The given function is $f\left(x\right)=x^2-4x-3$
$f$ being a polynomial function, is continuous in $[1,4]$ and is differentiable in $(1,4)$ and derivative is $f^{\prime }\left(x\right)=2x-4.$
Now $f\left(1\right)=1^2-4\cdot 1-3=-6,f\left(4\right)=4^2-4\cdot 4-3=-3$
$\therefore \frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(4\right)-f\left(1\right)}{4-1}=\frac{-3-(-6)}{3}=\frac{3}{3}=1$
Mean Value Theorem states that there is a point $c$ $\in (1,4)$ such that $f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}=1$
$\Rightarrow 2c-4=1$
$\Rightarrow c=\frac{5}{2}\in (1,4)$
Hence, Mean Value Theorem is verified for the given function.