In this problem, we're asked to determine where the local and global maxima and minima occur on the graph of our function f(x). We're given the graph of our sine function. At first glance, this might seem a bit tricky because there may seem to be many maxima and minima, since we know that the sine function oscillates up and down between 1 and negative one. Let's go ahead and dive in here.
Because we've been working with functions that just have one highest point and one lowest point, this may seem confusing. However, because all of these peaks and valleys occur at the same exact high point and the same exact low point, we actually have multiple points representing global maxima or global minima. Looking at this highest point at \( x = \frac{\pi}{2} \), we know that it reaches a value of 1. This is the highest point that this function ever reaches, indicating that this point is a global maximum.
This point is also a local maximum because it is higher than all nearby points, and it is not at the endpoint. Thus, it is both a global and local maximum. Now, looking over here at \( x = \frac{5\pi}{2} \), you might wonder whether this point, which also reaches 1, is also a global max. The answer is yes. Because the absolute highest value that this function reaches is 1, it's acceptable for it to achieve this value multiple times. This point, too, is both a global and local maximum.
The same principle applies to our minimum values. We know the absolute minimum value that the sine function reaches is negative one. Therefore, at \( x = \frac{3\pi}{2} \) and \( x = \frac{7\pi}{2} \), both points represent both a global and local minimum value. Dealing with these oscillating functions can be confusing, but it's understandable for a function to have multiple global maxima or minima since it reaches those values multiple times. It remains the highest or lowest point of the entire function. Feel free to let us know if you have any questions.
Let's keep going.
