Intervals of increase and decrease of a function

In this lesson I am going to explain how to calculate the intervals of increase and decrease of a function when we do not have its graph.

The intervals of increase and decrease of a function are also called monotony of a function.

How do we know at which intervals a function is increasing or decreasing?

We know whether a function is increasing or decreasing in an interval by studying the sign of its first derivative:

If the first derivative of the function f(x) is greater than zero at a point, then f(x) is strictly increasing at that point:

growth and decrease of a function solved exercises

If the first derivative of the function f(x) is less than zero at one point, then f(x) is strictly decreasing at that point:

intervals of growth and decrease of a function definition

Exercise solved on the calculation of intervals of growth and decrease of a function

For example, we are going to study the growth and decrease intervals of the next function:

growth and decrease intervals

We have to study the sign of its first derivative, therefore, the first thing to do is to calculate the first derivative of the function:

growth and decrease of a function

Now we are going to study its sign.

For it, we are going to calculate the roots of the function, that is to say, the values that make that the function is equal to 0. Therefore, we equal the function to 0 and solve it:

growth interval

Before I start solving, I simplify the coefficients:

intervals of growth and decrease of a function

As it is a third degree equation, I break it down into factors by the Ruffini rule:

growth interval and decrease

And I get their solutions, which are:

increasing and decreasing intervals resolved exercises

These three solutions are placed on the number line:

interval of growth and decrease of a function

We have 4 intervals left:

  • From least infinite to -2
  • From -2 to -1
  • From -1 to 1
  • From 1 to infinity

We need to know if f'(x) is positive or negative in each interval.

And how do we do that?

So we have to choose a value of x that belongs to each interval and calculate the value of f'(x) at that point.

For the interval that goes from least infinite to -2, I’m going to choose the point x=-3. Calculation f'(-3):

intervals of growth and decrease definition

f'(-3) is less than 0, then all points that are between less infinity and -2 will be negative and therefore the first derivative in that interval is less than zero, so f(x) is decreasing in that interval:

intervals of growth and decrease examples

This is how we represent it on the line:

intervals of a function

We continue with the interval that goes from -2 to -1. I choose the point x=-1,5 and calculate the value of f'(x) for that point:

definicion growth intervals

It is greater than zero, therefore, f(x) will be increasing in that interval:

Which are the growth and decrease intervals

And we represent it on the line:

growing interval

For the interval (-1,1) we choose the point x=0 and calculate f'(0):

growth and decrease intervals resolved exercises

It is less than zero, so f(x) will be decreased by (-1.1):

growth intervals

And we reflect it on the line:

definicion growth interval

For the last interval, we choose the point x=2 and calculate the value of the first derivative for x=2:

increasing and decreasing intervals of a function examples

Which is greater than zero, so f(x) will be increasing from 1 to infinity:

intervals of growth and decrease of a function solved exercises

And we leave it represented on the line:

intervals of growth of a function

We already have at what intervals the function is increasing and decreasing.

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