Concavity and the Second Derivative

Concave up
Concave down
The sign of the first derivative of a function gives us information about its monotonicity. The sign of the second derivative gives us information about its concavity.

If the second derivative of a function f(x) is defined on an interval (a,b) and f ''(x) > 0 on this interval, then the derivative of the derivative is positive. Thus the derivative is increasing! In other words, the graph of f is concave up. Similarly, if f ''(x) < 0 on (a,b), then the graph is concave down.

Conversely, if the graph is concave up or down, then the derivative is monotonic. Hence its derivative, i.e., the second derivative, does not change sign.

Concavity and The Second Derivative

Suppose that f is twice differentiable on the open interval (a,b).

If f is concave up [down] on (a,b) then f ''(x) > 0 [f ''(x) < 0] on (a,b).
If f ''(x) > 0 (f ''(x) < 0) on (a,b), then then f(x) is concave up [down] on the interval (a,b).

Inflection Points

Inflection points are points where the concavity changes sign. These are the points where the second derivative changes sign and so are easily recognized if we have information on the sign of the second derivative.

Applications of the Second Derivative

Just as the first derivative appears in many applications, so does the second derivative. Here are some questions which ask you to identify second derivatives and interpret concavity in context.

Instructions: For each of the following sentences, identify

A function whose second derivative is being discussed.
What is being said about the concavity of that function.

"Wall Street reacted to the latest report that the rate of inflation is slowing down."

"When he saw the light turn yellow, he floored it."

"As the immunization program took hold, the rate of new infections decreased dramatically."

Equipment Check: Use the second derivative to explain why the graph of sin(x) is concave down if and only if the value of the function is positive. Can you interpret this graphically?

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