Concavity and the Second Derivative
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
0 ( f ''(x) < 0 ) on (a,b), then then f(x) is concave up [down] on the interval (a,b) .
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.