# Given the function g(x) = x^3+9x^2+11, find: a. Critical points, b. Intervals where the function...

## Question:

Given the function {eq}g(x) = x^3+9x^2+11 {/eq}, find:

a. Critical points,

b. Intervals where the function is increasing,

c. Intervals where the function is decreasing,

d. Inflection points.

## Functions:

We can find the critical points of a function {eq}y=f(x) {/eq} by setting the first derivative i.e., {eq}y'=0 {/eq}. Hence we can calculate the increasing/decreasing nature using the monotone intervals and the inflection points can be calculated by setting {eq}f''(x)=0. {/eq}

## Answer and Explanation:

Given: {eq}g(x) = x^3+9x^2+11 {/eq}

a) Finding the critical points by setting {eq}g'(x) = 0 {/eq}

{eq}\Rightarrow 3x^2+18x=0\\\Rightarrow 3x(x+6)=0\\\Rightarrow x=0,-6 {/eq}

b) In the interval {eq}(-\infty,-6) \ , g'(x)>0\rightarrow {/eq} Increasing.

In the interval {eq}(0,\infty) \ , g'(x)>0\rightarrow {/eq} Increasing.

c) In the interval {eq}(-6,0) \ , g'(x)<0\rightarrow {/eq} Decreasing.

d) Finding the inflection points by setting {eq}f''(x)=0 {/eq}

{eq}\Rightarrow 6x+18=0\\\Rightarrow x=-3 {/eq}