Question

Verify Lagrange's mean value theorem for the function $f\left(x\right)=(x-1)(x-2)(x-3)$ in interval $(0,4)$ and also find out the value of $C$.

Answer 1

$f\left(x\right)=(x-1)(x-2)(x-3)$

$(0,4)$

According to Lagrange's mean value theorem

$f\left(o\right)=(-1)(-2)(-3)$ $f^{\prime }\left(o\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$f\left(4\right)=\left(3\right)\left(2\right)\left(1\right)$ $=\frac{6-(-6)}{4-0}=\frac{6+6}{4}=\frac{12}{4}$

$f\left(o\right)=3$

$f\left(o\right)=3x$