Question Video: Identifying the 𝑥-Coordinates of the Turning Points of a Function from a Graph of Its First Derivative Mathematics

Video Transcript

Use the given graph of a function 𝑓 prime to find the 𝑥-coordinates of all the points that have a local maximum or minimum of 𝑓.

The graph we’ve been given is the graph of the derivative of the function 𝑓 of 𝑥. And we’re asked to use this graph to determine the 𝑥-coordinates of all the points on the graph of 𝑓 itself, which are either local minima or local maxima. Let’s begin by recalling what we know about local minima and local maxima. They’re also called turning points or sometimes stationary points of a function. And they have a distinctive shape and a certain set of properties.

At a local maximum, the graph of the function will look like this. The slope of the function and, hence, its first derivative is equal to zero at the local maximum itself. And the graph changes from sloping upwards to sloping downwards. This means that the first derivative of the function is positive immediately to the left of the local maximum and negative immediately to the right.

For the local minimum, most of this is reversed. The slope of the function and, hence, its first derivative is still equal to zero at the local minimum itself. But this time, the graph changes from sloping downward to sloping upward. This means the first derivative of the function is negative immediately to the left of the local minimum and positive immediately to the right.

So, to identify local minima and local maxima of the function 𝑓 from the graph of its derivative, we need to consider two things. First, we need to consider where the first derivative is equal to zero. This will give us the 𝑥-coordinates of any turning points on the graph of 𝑓. Secondly, we need to consider what the sign of the first derivative is immediately either side of these 𝑥-values. The sign of the first derivative will enable us to distinguish between whether these points are local maxima or local minima, or they could indeed be points of inflection.

Let’s look at the graph of 𝑓 prime of 𝑥 then. We can see that it intersects the 𝑥-axis three times at 𝑥 equals one, 𝑥 equals six, and 𝑥 equals eight. So these are the 𝑥-coordinates of the turning points of 𝑓 of 𝑥. When 𝑥 equals one, the first derivative is negative immediately to the left of this value and positive immediately to the right of this value. This tells us that the graph of 𝑓 of 𝑥 has a local minimum at 𝑥 equals one. The first derivative is zero when 𝑥 equals one, and it changes from negative to positive. At 𝑥 equals eight, the same pattern of behavior is true. 𝑓 prime of 𝑥 is equal to zero, and it changes from negative to positive as we move from left to right. So we can say that the graph of 𝑓 has a local minimum at 𝑥 equals one and at 𝑥 equals eight.

At 𝑥 equals six, however, the derivative is positive immediately to the left of this value and negative immediately to the right. This tells us that the stationary point at 𝑥 equals six on the graph of 𝑓 is a local maximum. So we can state our answer. 𝑓 has a local minimum at 𝑥 equals one and 𝑥 equals eight and a local maximum at 𝑥 equals six.

Don’t be misled into looking for turning points on the graph we’ve been given. This graph has local maxima at 𝑥 equals two and 𝑥 equals five. And it has local minima at 𝑥 equals three and 𝑥 equals seven. But these are the turning points of the graph of the first derivative of 𝑓, not the graph of 𝑓 itself. Our answer is that 𝑓 has a local minimum at 𝑥 equals one and 𝑥 equals eight and a local maximum at 𝑥 equals six.