Question:
Verify Mean Value Theorem, if $f(x)=x^{2}-4 x-3$ in the interval $[a, b]$, where $a=1$ and $b=4$.
Solution:
The given function is $f(x)=x^{2}-4 x-3$
f, being a polynomial function, is continuous in [1, 4] and is differentiable in (1, 4) whose derivative is 2x − 4.
$f(1)=1^{2}-4 \times 1-3=-6, f(4)=4^{2}-4 \times 4-3=-3$
$\therefore \frac{f(b)-f(a)}{b-a}=\frac{f(4)-f(1)}{4-1}=\frac{-3-(-6)}{3}=\frac{3}{3}=1$
Mean Value Theorem states that there is a point $c \in(1,4)$ such that $f^{\prime}(c)=1$
$f^{\prime}(c)=1$
$\Rightarrow 2 c-4=1$
$\Rightarrow c=\frac{5}{2}$, where $c=\frac{5}{2} \in(1,4)$
Hence, Mean Value Theorem is verified for the given function.