### Video Transcript

The graph of π is shown. Which of the following is true about the function at π₯ equals three. a) π prime of three is undefined. b) π prime of three is zero. c) π of three is undefined. Or d) None of the other choices are correct.

In this question, weβve been given a graph of a function π and asked to evaluate four statements. Two of the statements are about π prime of three. Thatβs the derivative of our function π evaluated at three. One of the statements is about the nature of the function itself at π₯ equals three. And the fourth statement is about whether the other statements are true.

Weβll begin by looking at this third statement; π of three is undefined. Letβs find π of three on our graph. π₯ equals three is here. We can see that our graph of our function π has a point with coordinates three, zero. There are certainly no discontinuities on our graph. We can therefore say that π of three is not undefined. In fact, itβs zero. So this third statement canβt be true. So what about the first two statements? This is about π prime, the derivative of our function. Remember, for a function to be differentiable, we need to be able to find the derivative of that function at a point on the graph. Here, thatβs at the point where π₯ is equal to three.

Now, we have a little bit of a problem. We can see that the slope of our graph and therefore the derivative as it approaches π₯ equals three from the left is positive. And the slope of our graph and therefore the derivative as it approaches π₯ equals three from the right is negative. Since the left- and right-hand limits of the derivative disagree as π₯ approaches three, we conclude that we cannot evaluate the derivative of the function at this point. And we say that π prime of three is undefined.

And the answer is a. In fact, when a graph has a sharp turn, as our graph has at π₯ equals three, the derivative is indeed undefined at that point.