Question

Verify LMVT for $f\left(x\right)=(x-1)(x-5)(x+10)$ in the interval $[0,5]$.

Answer 1

$f\left(x\right)=(x-1)(x-5)(x+10)$

$f^{\prime }\left(x\right)=(x-5)(x+10)+(x-1)(x+10)+(x-1)(x-5)$

$=x^2+5x-50+x^2+9x-10+x^2-6x+5$

$=3x^2+8x+55$

$\frac{f\left(5\right)-f\left(0\right)}{5-0}=3c^2+8c+55$

$\frac{(5-1)(5-5)(5+10)-(0-1)(0-5)(0+10)}{5-0}=3c^2+8c+55$

$-\frac{50}{5}=3c^2+8c+55$

$3c^2+8c+65=0$

$c=\frac{-8\pm \sqrt{8^2-4\times 3\times 65}}{2\times 8}$

$c=\frac{-4\pm i\sqrt{179}}{3}$