find all critical number f(x)=((x+1)^(1/3) (2 X+5)^3)

Question

Find all of the critical numbers of f if f(x)=(x+1)^(1/3) (2 X+5)^3

Answer 1

f(x)=(x+1)^⅓(2x+5)³.

f'(x)=⅓(x+1)^-⅔(2x+5)³+6(x+1)^⅓(2x+5)²=0 at critical points.

Multiply through by 3(x+1)^⅔:

(2x+5)³+18(x+1)(2x+5)²=0,

(2x+5)²(2x+5+18x+18)=0,

(2x+5)²(20x+23)=0, so x=-2.5 or -23/20=-1.15. These are the critical numbers.

f(-2.5)=0; f(-1.15)=(∛-0.15)(2.7)³=-10.4582 approx.

(The point (-2.5,0) is a point of inflection (neither max or min), while (-1.15,-10.4582) is a minimum.)