Use the given graph of 𝑓 to find all possible intervals on which 𝑓 is concave downward.
And then we have the graph of 𝑓 plotted within the first quadrant. So let’s begin by reminding ourselves what it means for a function 𝑓 to be concave downward. Let’s begin by thinking about the algebraic definition. Given a graph of 𝑓, it will be concave downward if 𝑓 prime is decreasing, in other words, if the rate of change of the slope of the function is decreasing.
But what does this mean for the graph of the function? How can we quickly identify the intervals on which the graph is conclave downward? To think about this graphically, we need to think about what the tangent lines to the curve look like. If 𝑓 prime is decreasing, the graph of 𝑓 is concave downwards and the tangent lines will all lie above the curve of 𝑓 at a given point.
For instance, let’s take the very first portion of our graph. We can draw the tangents at approximately 𝑥 equals 0.5, 𝑥 equals one, and 𝑥 equals 1.5. We see that the tangents all lie above the curve of our function 𝑓. In particular, we notice that the slope of each line must be decreasing. So this first part is certainly concave downwards.
But where does it stop? Well, we notice that if we attempt to draw the tangent lines to the curve at approximately 𝑥 equals 2.4, 𝑥 equals three, and 𝑥 equals 3.3, the tangent lines lie below the curve of the function. So at this point, 𝑓 is actually concave upward. We might observe that the change over point here is at 𝑥 equals two. So the curve is concave downwards between 𝑥 equals zero and 𝑥 equals two.
But is it concave downwards at any other locations? Well, yes. The slope of 𝑓 is actually decreasing over a second interval. Specifically, this appears to happen at 𝑥 equals four, and it stops at 𝑥 equals five, where the slope is in fact zero. Since 𝑓 prime is neither decreasing nor increasing at points where it’s equal to zero, then the second interval is from four to five.
And so we’ve identified the intervals on which 𝑓 is concave downward. Now to save us time in future, we can observe that when 𝑓 is concave downwards, we have something that looks a little bit like a cave. When it’s concave upward, we have something that looks like a cup. So 𝑓 is concave downward on the open interval from zero to two and the open interval from four to five.