The graph of 𝑓 is shown. Which of the following is true about the function at 𝑥 equals three. a) 𝑓 prime of three is undefined. b) 𝑓 prime of three is zero. c) 𝑓 of three is undefined. Or d) None of the other choices are correct.
In this question, we’ve been given a graph of a function 𝑓 and asked to evaluate four statements. Two of the statements are about 𝑓 prime of three. That’s the derivative of our function 𝑓 evaluated at three. One of the statements is about the nature of the function itself at 𝑥 equals three. And the fourth statement is about whether the other statements are true.
We’ll begin by looking at this third statement; 𝑓 of three is undefined. Let’s find 𝑓 of three on our graph. 𝑥 equals three is here. We can see that our graph of our function 𝑓 has a point with coordinates three, zero. There are certainly no discontinuities on our graph. We can therefore say that 𝑓 of three is not undefined. In fact, it’s zero. So this third statement can’t be true. So what about the first two statements? This is about 𝑓 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, that’s at the point where 𝑥 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 𝑥 equals three from the left is positive. And the slope of our graph and therefore the derivative as it approaches 𝑥 equals three from the right is negative. Since the left- and right-hand limits of the derivative disagree as 𝑥 approaches three, we conclude that we cannot evaluate the derivative of the function at this point. And we say that 𝑓 prime of three is undefined.
And the answer is a. In fact, when a graph has a sharp turn, as our graph has at 𝑥 equals three, the derivative is indeed undefined at that point.