Video Transcript
Which of the following statements correctly describes the monotonicity of the function represented in the figure below? Is it (A) the function is increasing on five, eight; constant on negative one, five; and decreasing on negative two, negative one? (B) The function is increasing on negative two, negative one; constant on negative one, five; and decreasing on five, eight. (C) The function is increasing on negative two, five and decreasing on five, eight. Or (D) the function is increasing on five, eight and decreasing on negative two, five.
By reading the question, we’ve probably inferred what we mean by the monotonicity of a function. This simply tells us if a function is increasing or decreasing. We recall that if a function is increasing over some interval, it has a positive slope. And if the function is decreasing, it has a negative slope. Finally, if the function is constant, it is represented by a horizontal line. So let’s consider the graph of our function.
We see that it has three main sections. The first section is between negative two and negative one. The next section is between negative one and five, whilst the third section is between five and eight on the 𝑥-axis. So let’s consider each section in turn. We can see that the slope of the first part of our function must be positive. It’s sloping upwards. We then have a horizontal line between 𝑥 equals negative one and 𝑥 equals five. And the third part of our graph has a negative slope. It’s sloping downwards. Our function is therefore increasing for some time, it’s constant, and then finally it’s decreasing. We need to decide the intervals over which each of these occur.
It has a positive slope between 𝑥 equals negative two and negative one. And so we define this using the open interval negative two to negative one. We are not going to use a closed interval as we don’t really know what’s happening at the endpoints of the interval. For example, when 𝑥 is equal to negative one, the graph of our function has this sort of sharp corner. As a result, we’re going to leave 𝑥 equals negative two and 𝑥 equals negative one out of our interval.
In a similar way, the function is constant over the open interval negative one to five. And it’s decreasing over the open interval five to eight. Once again, we don’t know what’s really happening at the endpoints, but we do have sharp corners. And as such, we can’t say whether it’s increasing, decreasing, or constant.
We can therefore conclude that the correct answer is (B). The function is increasing on the open interval from negative two to negative one, constant on the open interval from negative one to five, and decreasing on the open interval from five to eight.
