Increasing and decreasing functions. Find the intervals on whi...

Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = cos² x on [-π,π]

Accepted Answer

Hello, in this video, we are going to be determining the intervals of increasing and decreasing. We want to determine the intervals on which the function F X equal to square of X is increasing or decreasing on the interval from negative pi divided by 2 to pi divided by 2. In order to determine the intervals of increasing and decreasing, the first thing we need to do is we need to obtain the critical points to the function. In order to obtain the critical points, we first need to calculate the derivative. So, the function given to us is F of X equal to sin squared of X. We will take the derivative with respect to X by using the chain rule. The derivative of sine squared X will be 2 sin of x cosine of X. Next, notice that we have a Pythagorean identity. To sin of x cosine of X is the Pythagorean identity to sine double angle of X. So we can rewrite the derivative as sign of 2 X. And this is going to be the derivative of the given function. Now that we have the derivative, in order to obtain the critical points, we want to solve for the values of X that cause the derivative to be zero. So we will take sine of 2 X and set it equal to 0. Now, keep in mind, we want to solve for the values of X on the interval from negative pi divided by 2 to pi divided by 2. This is describing the right the right hand side of the unit circle. So, on the right hand side of the unit circle in the interval, what's the value of sign that is going to be zero? Well, since sine represents the Y values of any coordinate along the unit circle, the only value of Y that is equal to 0 is going to be located at the angle of 0. What this means is that there is only one critical point to the derivative or to the original function. That critical point is X is equal to 0. Now that we have the critical points and the boundaries of our interval, we will go ahead and create a number line. Now, on this number line, we are going to label the values of our boundaries. We have negative pi divided by 2, and positive pi divided by 2. Then we will label our critical point, which is located at 0. What we are going to do now is we are going to pick testing values within the intervals to the left and right hand side of 0. We will take those values and plug it into the derivative. If the derivative is negative, then the graph is decreasing on that interval, and if it is positive, the graph is increasing on that interval. So, what's a testing point that we can pick on the interval from negative pi divided by 2 to 0? Well, a testing point we can pick is negative pi divided by 4. If we take the value negative pi divided by 4 and plug it into the derivative, we have F of negative pi divided by 4 equal to sine of 2 multiplied by negative pi divided by 4. The argument of sine will simplify to be sine of negative pi divided by 2. Now, since sine is an odd function, we can rewrite sine of negative pi divided by 2 to be negative sign of pi divided by 2. This will further simplify to be negative 1. Since the value at negative pi divided by 4 is a negative number, that tells us that the graph of F of X will be decreasing on the interval from negative pi divided by 2 to 0. Now, we will go ahead and repeat this process with the interval from 0 to pi divided by 2. We will pick the testing point positive pi divided by 4. If we take this testing point and plug it into the derivative, we have F of pi divided by 4, equal to sin of 2 multiplied by pi divided by 4. This will simplify to be sin of pi divided by 2, which will further simplify to be positive 1. And since the value of this testing point is positive, that tells us that the graph F of X is increasing on the interval from 0 to pi divided by 2. So, to summarize, we calculated that the interval of increase of the original function F of X is from 0 to pi divided by 2. And the interval of decreasing for the original function F of X is from negative pi divided by 2 to 0, and this is going to be the solution to this problem. So I hope this video helps you in understanding how to identify the intervals of increasing and decreasing, and I will go ahead and see you all in the next video.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = cos² x on [-π,π]

Key Concepts

Derivative

Critical Points

First Derivative Test

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