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:
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:
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:
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:
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:
Before I start solving, I simplify the coefficients:
As it is a third degree equation, I break it down into factors by the Ruffini rule:
And I get their solutions, which are:
These three solutions are placed on the number line:
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):
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:
This is how we represent it on the line:
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:
It is greater than zero, therefore, f(x) will be increasing in that interval:
And we represent it on the line:
For the interval (-1,1) we choose the point x=0 and calculate f'(0):
It is less than zero, so f(x) will be decreased by (-1.1):
And we reflect it on the line:
For the last interval, we choose the point x=2 and calculate the value of the first derivative for x=2:
Which is greater than zero, so f(x) will be increasing from 1 to infinity:
And we leave it represented on the line:
We already have at what intervals the function is increasing and decreasing.
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