Video Transcript
Suppose that 𝑓 is a continuous function and is differentiable everywhere. Suppose also that 𝑓 of three equals negative two, 𝑓 of six equals three, and 𝑓 of negative seven equals negative four. Which of the following statements must be true about 𝑓? One, 𝑓 has exactly two zeros. Two, 𝑓 has at least two zeros. Three, 𝑓 must have a zero between three and six. Four, there is not enough information to determine anything about the zeros of 𝑓.
First, let’s remind ourselves what is meant by the word zero or zeros in this context. The zeros of a function are the 𝑥 values at which its graph either crosses or touches the 𝑥-axis indicated with orange dots in the sketch on the right. We’ll use the information given in the question which is the values of the function for three values of 𝑥 to determine something about its graph’s behavior. We’ll draw a very rough sketch of the functions or at least of some points on the function. And we’ll just consider the key 𝑥 values of three, six, and negative seven. We’re told, first of all, that 𝑓 of three is equal to negative two. So there’s a point somewhere around here which lies on the graph of our function.
We’re also told that 𝑓 of six is equal to three. So there’s a point somewhere around here which is also on the graph of our function. Finally, we’re told that 𝑓 of negative seven is equal to negative four. So there’s also a point somewhere around here. Now, we don’t have any other information about the shape of the curve or its behavior between these points. But we do know that 𝑓 is a continuous function and is differentiable everywhere. This tells us that there are no gaps or discontinuities in the graph of our function 𝑓. That means that these three points are all joined together with a continuous line. It may look something like the line I’ve sketched in pink. Or it may look something like the curve I have sketched in orange. We don’t know. But we do know that there will be a continuous and differentiable curve connecting these three points.
Now remember, the statements we were given concern the zeros of this function 𝑓. So let’s have a look at what we do know. We know that 𝑓 of three is negative and 𝑓 of six is positive. And as there is a continuous curve connecting these two points, we know that this curve must cross the 𝑥-axis somewhere between the 𝑥-values of three and six. Those are the points that I have now identified in green on our pink and orange curves. This tells us that our function 𝑓 must have a zero between three and six. The value of the function can’t change from negative to positive without going through zero. Now, let’s look at the other statements.
Well, we can certainly rule out statement four. There is enough information for us to determine something about the zeros of 𝑓. We know that 𝑓 has a zero between three and six. However, we don’t know anything else about the behavior of this curve. We have sketched one example in orange which has three zeros. So that would rule out statement one. And we’ve sketched one example in pink which has just one zero. So that would rule out statement two. And remember, these are just two examples. There are many other possibilities for the graph of 𝑓.
So our answer to the question, which of the following statements must be true about 𝑓, is statement three. 𝑓 must have a zero between three and six because the sign of the function changes from negative to positive between these values. And therefore, the function must cross the 𝑥-axis somewhere between three and six.
