Question

How do you find the extrema and points of inflection for `f(x) =(3x)/((x+8)^2)`?

Answer 1

Derivate once :

`((3x)/(x+8)^2)' = (3(x+8)^2-3x(2(x+8)))/(x+8)^4 = -(3(x-8))/(x+8)^3`

Root is `x=8` We have global maxima at `x = 8` and `y = 3/32`

Derivate again `-(3(x-8))/(x+8)^3` = `-(3(x+8)^3-9(x-8)(x+8)^2)/(x+8)^6 = (6(x-16))/(x+8)^4`

Root is `x = 16` We have a points of inflection at `x = 16` and `y = 1/12`