Verify Mean Value Theorem, if $f(x)=x^{2}-4 x-3$ in the interval $[a, b],$ where $a=1$ and $b=4$
Verify Mean Value Theorem, if $f(x)=x^{2}-4 x-3$ in the interval $[a, b],$ where $a=1$ and $b=4$
The given function is $f(x)=x^{2}-4 x-3$
$f,$ being a polynomial function, is a continuous in $[1,4]$ and is differentiable in $(1,4)$ whose derivative is $2 x-4$
$f(1)=1^{2}-4 \times 1-3=6, f(4)=4^{2}-4 \times 4-3=-3$
$\therefore \frac{f(b)-f(a)}{b-a}=\frac{f(4)-f(1)}{4-1}=\frac{-3-(-6)}{3}=\frac{3}{3}=1$
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 foer the given function.
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