Question

Find the points of maximum and minimum and the intervals of monotonicity of the following functions.
$f\left(x\right)=\frac{(x-2)(8-x)}{x^2}$

Answer 1

$f\left(x\right)=\frac{-x^2+10x-16}{x^2}$

$f^{\prime }\left(x\right)=\frac{(-2x+10)x^2-2x(-x^2+10x-16)}{x^4}=\frac{-10x^2+32x}{x^4}=0$

$x=0,3.2$

$f\left(0\right)=\infty$ is maximum

$f\left(3.2\right)=0.5625$ is minimum