The graph of 𝑓 prime, the derivative of 𝑓, is shown in the figure. At which 𝑥-value does the graph of 𝑓 have a point of inflection?
Let’s begin by recalling what a point of inflection is. We can recognize points of inflection as points on a curve at which there is a change in the direction of the curvature, either from concave to convex or vice versa. The gradient of the curve, the function’s first derivative, also has the same sign either side of the point of inflection.
Now, some points of inflection are also stationary or critical points of a curve; that is, points where the first derivative of the function is equal to zero. However, points of inflection do not have to be stationary points. It’s possible to have a point of inflection that is a change in direction of curvature without the first derivative being equal to zero.
It’s also true that the sign of the second derivative — that’s 𝑓 double prime — changes around the point of inflection, either from positive to negative or from negative to positive. But at the point of inflection itself, the second derivative 𝑓 double prime of 𝑥 is equal to zero. This will help us with determining at which of the four 𝑥-values, the graph of 𝑓 has a point of inflection. Remember that the graph we’ve been given is the graph of 𝑓 prime, the first derivative of the function.
The second derivative is the derivative of the first derivative, which means that if we want to identify points where the second derivative is equal to zero, we need to look for points on the graph of the first derivative where the gradient of the tangent to the curve is equal to zero. The gradient or derivative of the first derivative gives the second derivative.
We can see that if we draw tangents to the graph of 𝑓 prime at 𝑥 one, 𝑥 three, and 𝑥 four, these do not have a gradient of zero. However, if we draw a tangent to the graph of 𝑓 prime at 𝑥 two, we can see that it does have a gradient of zero because it is a horizontal line which tells us that 𝑓 double prime of 𝑥 two is equal to zero.
We also note that the sign of the first derivative 𝑓 prime is the same either side of 𝑥 two because the graph is above the 𝑥-axis. So 𝑓 prime is positive to the left of 𝑥 two and 𝑓 prime is positive to the right of 𝑥 two.
So as the second derivative of 𝑓 is equal to zero at 𝑥 two which we saw by drawing a tangent to the graph of the first derivative at this point, we can conclude that the 𝑥-value at which the graph of 𝑓 has a point of inflection is 𝑥 two.