Video Transcript
Which graph has three real zeros and two local maxima?
Let’s begin by recalling the definitions of the zeros of a graph and local maxima. Firstly, the zeros of a function 𝑓 of 𝑥 are the values of 𝑥 for which the function 𝑓 of 𝑥 is equal to zero. On a graph of that function, the zeros are its 𝑥-intercepts, the 𝑥-values at which the graph either crosses or touches the 𝑥-axis. Considering graph (a) first of all, we can see that it intercepts the 𝑥-axis in four places. And so the function shown in graph (a) has four real zeros.
Looking at graph (b), we can see that it intercepts the 𝑥-axis twice and it touches the 𝑥-axis once. And so graph (b) has three real zeros. Graph (c) intercepts the 𝑥-axis in three places, and so graph (c) also has three real zeros. Notice that the question mentions three real zeros. It’s possible that there may be other values of 𝑥 for which any of these functions is equal to zero. But if these are nonreal values, then we aren’t concerned. We’re looking for three real zeros, and so we’ve narrowed our options down to (b) and (c).
Next, let’s recall what we mean by local maxima. A local maximum is a point on the graph of a function at which two things are true. Firstly, the first derivative or slope of the function, 𝑓 prime of 𝑥, is equal to zero. And the slope changes from positive to negative as the 𝑥-values increase. This means that the shape of the graph of the function around this value will look like this. This is called a local maximum because the function at that particular point is at its largest value in the region immediately surrounding that point.
Looking at graph (b) first of all then, we can identify two local maxima at these points here. In graph (c), however, we only identify a single local maximum. For completeness, let’s just observe that there are two local maxima in graph (a). Graph (b) is therefore the only graph which has the right number of real zeros and the right number of local maxima. And so our answer is (b).
