## The Second Derivative and the Shape of a Function f(x)

Read this section to learn how the second derivative is used to determine the shape of functions. Work through practice problems 1-9.

### f'' and Extreme Values

The concavity of a function can also help us determine whether a critical point is a maximum or minimum or neither. For example, if a point is at the bottom of a concave up function (Fig. 7), then the point is a minimum.

**Fig. 7**

**The Second Derivative Test for Extremes:**

(a) If and then is concave down and has a local maximum at .

(b)
If and then is concave up and has a local minimum at .

(c) If and
then may have a local maximum, a minimum or neither at .

Proof: (a) Assume that . If then is concave down at so the graph of will be below the tangent line
for values of near . The tangent line, however, is given by , so if is close to then and has a local maximum at . The proof
of (b) for a local minimum of is similar.

(c) If and , then we cannot immediately conclude anything about local maximums or minimums of at . The functions , and
all have their first and second derivatives equal to zero at , but has a local minimum at has a local maximum at , and has neither a local maximum nor a local minimum at .

The Second Derivative Test for Extremes is very useful when is easy to calculate and evaluate. Sometimes, however, the First Derivative Test or simply a graph of the function is an easier way to determine if we have a local
maximum or a local minimum – it depends on the function and on which tools you have available to help you.

**Practice 2:** has critical numbers and . Use the Second Derivative Test for Extremes to determine whether and
are maximums or minimums or neither.