Question

c) Determine the location and values of the absolute maximum and absolute minimum for the given function: 𝑓(𝑥) = (−𝑥 + 2) ସ , 𝑤ℎ𝑒𝑟𝑒 0 ≤ 𝑥 ≤ 3

Answer 1

$$F(x)=(-x+2)^3$$

$0\leq x\leq 3$

Find the first derivative

$$F'(x)=((-x+2)^3)'=-3(-x+2)^2$$

Find the critical number(s)

$$F'(x)=0=>-3(-x+2)^2=0$$ $$x=2$$

Differentiate f'(x) w.r.t. x-

F''(x)=6(-x+2)

F''(2)=6(-2+2)=0

So F(x) has neither maximum nor minimum at x=2.

$F(0)=(-0+2)^3=8\\ F(3)=(-3+2)^3=-1\\ F(2)=(-2+2)^3=0$

So F(x) has absolute maximum at x=0 and The value is 8.

F(x) has absolute minimum at x=3 and The value is -1.