2.6: Second Derivative and Concavity 

Second Derivative and Concavity

Graphically, a function is concave up if its graph is curved with the opening upward (Figure 1a). Similarly, a function is concave down if its graph opens downward (Figure 1b).

This figure shows the concavity of a function at several points. Notice that a function can be concave up regardless of whether it is increasing or decreasing.

For example, an epidemic : Suppose an epidemic has started, and you, as a member of congress, must decide whether the current methods are effectively fighting the spread of the disease or whether more drastic measures and more money are needed. In Figure 2 below, [latex]f(x)[/latex] is the number of people who have the disease at time [latex]x[/latex], and two different situations are shown. In both Figure 2a and Figure 2b, the number of people with the disease, [latex]f(\text{now})[/latex], and the rate at which new people are getting sick, [latex]f'(\text{now})[/latex], are the same. The difference in the two situations is the concavity of [latex]f[/latex], and that difference in concavity might have a big effect on your decision.

In Figure 2a, [latex]f[/latex] is concave down at “now”, the slopes are decreasing, and it looks as if it is tailing off. We can say “[latex]f[/latex] is increasing at a decreasing rate.” It appears that the current methods are starting to bring the epidemic under control.

In Figure 2b, [latex]f[/latex] is concave up, the slopes are increasing, and it looks as if it will keep increasing faster and faster. It appears that the epidemic is still out of control.

The differences between the graphs come from whether the derivative is increasing or decreasing.

The derivative of a function f is a function that gives information about the slope of [latex]f[/latex]. The derivative tells us if the original function is increasing or decreasing .

Because [latex]f'[/latex] is a function, we can take its derivative. This second derivative also gives us information about our original function [latex]f[/latex]. The second derivative gives us a mathematical way to tell how the graph of a function is curved. The second derivative tells us if the original function is concave up or down.

Second Derivative

Consider a function [latex]f(x)[/latex].

The second derivative of [latex]f[/latex] is the derivative of [latex]f'(x)[/latex].

Using prime notation, this is [latex]f''(x)[/latex]. You can read this aloud as f double prime of x.

Using Leibniz notation, the second derivative is written [latex]\frac{d^2y}{dx^2}[/latex]. This is read aloud as the second derivative of f with respect to x.

Derivatives and the Graph of a Function

The first derivative tells us if a function is increasing or decreasing.

The second derivative tells us if a function is concave up or concave down

You are likely experiencing the second derivative every day! If [latex]f(x)[/latex] represents the position of a particle at time [latex]x[/latex], then [latex]v(x) = f '(x)[/latex] will represent the velocity (rate of change of the position) of the particle and [latex]a(x) = v '(x) = f ''(x)[/latex] will represent the acceleration (the rate of change of the velocity) of the particle.

You are probably familiar with acceleration from driving or riding in a car. The speedometer tells you your velocity (speed). When you leave from a stop and press down on the gas pedal, you are accelerating – increasing your speed. The curve that would describe your position  over that time period would be concave up. However, if you were slowing down to a stop, you would be decreasing your speed, and the curve describing your position while you are slowing down would be concave down.

Let’s consider an example in business.

Inflection Points