Question

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

Consider the function $f(x)=(-x+2)^3.$

$Df:(-\infin, \infin).$

Find the first derivative with respect to $x$

$$f'(x)=(-x+2)^3=3(-x+2)^2(-1)$$

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

Find the critical number(s):

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

$$x=2$$

Critical number: $2.$

If $0\leq x\leq 3$

$$f(0)=(-0+2)^3=8$$

$$f(3)=(-3+2)^3=-1$$

$$f(2)=(-2+2)^3=0$$

The function $f(x)$ has the absolute maximum with value of $8$ on $[0,3]$ at $x=0.$

The function $f(x)$ has the absolute minimum with value of $-1$ on $[0,3]$ at $x=3.$