Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = x3 − 6x2 − 15x + 2 (a) Find the interval on which f is increasing. (Enter your answer using interval notation.) Find the interval on which f is decreasing. (Enter your answer using interval notation.)

a)

b)

Step-by-step explanation:

The first step to solve this question is finding the roots of the derivative of x.

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

.

This polynomial has roots such that , given by the following formulas:

In this question:

So

Finding the roots:

Simplifying by -3

So

Then

So the function can be divided in three intervals.

They are:

Less than -1

Between -1 and 5

Higher than 5

In which it increases and which it decreases?

Less than -1

Lets find the derivative in a point in this interval, for example, -2

Positive.

So in the interval of , the function increases.

Between -1 and 5

Will choose 0.

Negative.

So in the interval of , the function decreases.

Higher than 5

Will choose 6.

Positive

So in the interval of , the function increases.

(a) Find the interval on which f is increasing.

Using interval notation

b) Find the interval on which f is decreasing.