Question

 Consider the function f(x)=7(x−4)^2/3

f(x)=7(x−4)2/3. For this function there are two important intervals: (−∞,A)

(−∞,A) and (A,∞)

(A,∞) where A

A is a critical number.

Answer 1

$$f(x)=7(x-4)^{2/3}$$

Domain: $(-\infin, \infin)$

$$f'(x)=(7(x-4)^{2/3})' =7(\dfrac{2}{3})(x-4)^{-1/3}=\dfrac{14}{3}(x-4)^{-1/3}$$

$$=\dfrac{14}{3}(x-4)^{-1/3}$$

Find the critical number(s)

$f'(x)$ does not exist, if $x=4.$

The critical number is $4.$

If $x\in (-\infin, 4), f'(x)<0, f(x)$ decreases.

If $x\in (4,\infin), f'(x)>0, f(x)$ increases.