Question

Given the function g(x)=(x+7^3

a) Find the critical points of g(x)

b) On what intervals is g(x) increasing and decreasing

c) At what points, if any does g(x) local and absolute minimum and maximum values?

Answer 1

$$g(x)=(x+7)^3$$

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

1.

Find the first derivative with respect to $x$

$$g'(x)=((x+7)^3)'=3(x+7)^2$$

Find the critical number(s)

$$g'(x)=0=>3(x+7)^2=0=>x=-7.$$

Critical number: -7.

If $x<-7, g'(x)>0, g(x)$ increases.

If $x>-7, g'(x)>0, g(x)$ increases.

The critical point $x=-7.$

b)

The function $g(x)$ increases on $(-\infin, \infin).$

The function $g(x)$ is never decreasing.

c)

The function $g(x)$ is always increasing on $(-\infin, \infin).$

The function $g(x)$ has neither maximum nor minimum at $x=-7.$

The function $g(x)$ has no absolute maxium value on $(-\infin, \infin).$

The function $g(x)$ has no absolute minium value on $(-\infin, \infin).$