Video Transcript
Given that 𝑓 of 𝑥 is equal to sin two 𝑥 plus cos two 𝑥 on the interval 𝑥 is greater than or equal to zero and less than or equal to 𝜋, determine the intervals on which 𝑓 is increasing or decreasing.
We can work out whether a function is increasing or decreasing by differentiating the function. If 𝑓 prime of 𝑥 is greater than zero, then 𝑓 of 𝑥 is increasing. Likewise, if 𝑓 prime of 𝑥 is less than zero, then the function 𝑓 of 𝑥 is decreasing. The points at which 𝑓 prime of 𝑥 is equal to zero are the stationary or turning points. Let’s now consider the function in this question: sin two 𝑥 plus cos two 𝑥.
In order to differentiate this function, we will differentiate each term individually. The derivative of sin 𝑎𝑥 is 𝑎 cos 𝑎𝑥. When we differentiate sin two 𝑥, we get two cos two 𝑥. The derivative of cos 𝑎𝑥 is negative 𝑎 sin 𝑎𝑥. This means that differentiating cos two 𝑥 gives us negative two sin two 𝑥. Our next step is to work out the stationary or tuning points by setting 𝑓 prime of 𝑥 equal to zero. Two cos two 𝑥 minus two sin two 𝑥 is equal to zero. We can divide all three of these terms by two. Our equation simplifies to cos two 𝑥 minus sin two 𝑥 is equal to zero.
At this point, we recall that sin 𝑎𝑥 divided by cos 𝑎𝑥 is equal to tan 𝑎𝑥. We can divide both sides of the equation by cos two 𝑥. cos two 𝑥 divided by cos two 𝑥 is equal to one. Negative sin two 𝑥 divided by cos two 𝑥 is negative tan two 𝑥. Zero divided by cos two 𝑥 is equal to zero. Adding tan two 𝑥 to both sides of this equation gives us one is equal to tan two 𝑥 or tan two 𝑥 is equal to one.
Our next step is to solve this equation for all values between zero and 𝜋. Taking the inverse tan of both sides of the equation gives us two 𝑥 is equal to tan to the minus one or inverse tan of one. Ensuring that our calculator is in radian mode, this gives us an answer of 𝜋 over four. We need to calculate all the values of 𝑥 between zero and 𝜋. This means that two 𝑥 can be between zero and two 𝜋. In order to find our other solution, we could either draw the tan graph between zero and two 𝜋 or use a cast diagram, as shown.
As our value of tan two 𝑥 was positive, there will be one solution in the first quadrant and one solution in the third quadrant. We have already worked out that our first solution is 𝜋 over four. The second solution in the third quadrant will be equal to 𝜋 plus 𝜋 over four. This is equal to five 𝜋 over four. Whilst there are an infinite number of solutions, there are only two solutions in the range zero to two 𝜋. We can, therefore, say that two 𝑥 is equal to 𝜋 over four or five 𝜋 over four. Dividing both of these numbers by two gives us values of 𝑥 of 𝜋 over eight and five 𝜋 over eight. These are the two values between zero and 𝜋, where 𝑓 prime of 𝑥 is equal to zero.
We can now substitute in points slightly below and slightly above these values to work out whether 𝑓 prime of 𝑥 is increasing or decreasing around these points. Let’s firstly substitute a value slightly smaller than 𝜋 over eight. We’ll use 𝜋 over nine. At this point, 𝑓 prime of 𝑥 is equal to two cos two 𝜋 over nine minus two sin two 𝜋 over nine. This gives us a positive answer. When 𝑥 is equal to 𝜋 over nine, 𝑓 prime of 𝑥 is greater than zero. When we substitute 𝑥 equals 𝜋 over seven, 𝑓 prime of 𝑥 is negative. This means that it is less than zero. We can, therefore, conclude that when 𝑥 is 𝜋 over eight, we have a maximum point. The gradient 𝑓 prime of 𝑥 is increasing prior to this point and decreasing after this point.
We can now repeat this process with our second value five 𝜋 over eight. When 𝑥 is equal to five 𝜋 over nine, 𝑓 prime of 𝑥 is less than zero. When 𝑥 is equal to five 𝜋 over seven, 𝑓 prime of 𝑥 is greater than zero. This means that the point at which 𝑥 is five 𝜋 over eight is a minimum point, as the gradient 𝑓 prime of 𝑥 is decreasing before the point and increasing after the point. We can, therefore, say that the function 𝑓 of 𝑥 is increasing between zero and 𝜋 over eight and also between five 𝜋 over eight and 𝜋. The function is decreasing between 𝜋 over eight and five 𝜋 over eight as 𝑓 prime of 𝑥 in this interval is negative.
Sin two 𝑥 plus cos two 𝑥 increases between zero and 𝜋 over eight and also between five 𝜋 over eight and 𝜋, but decreases between 𝜋 over eight and five 𝜋 over eight.
