## 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.

### Inflection Points

**Definition:** An **inflection point** is a point on the graph of a function where the concavity of the function changes, from concave up to down or from concave down to up.

**Practice 3:** Which of the labelled points in Fig. 8 are inflection points?

**Fig. 8**

To find the inflection points of a function we can use the second derivative of the function. If , then the graph of is concave up at the point so is not an inflection point. Similarly,
if , then the graph of is concave down at the point and the point is not an inflection point. The only points left which can possibly be inflection points are the
places where is or undefined ( is not differentiable). To find the inflection points of a function we only need to check the points where is or
undefined. If or is undefined, then the point **may** or **may not** be an inflection point – we would need to check the concavity
of on each side of . The functions in the next example illustrate what can happen.

**Example 2:** Let and (Fig. 9). For which of these functions is the point an inflection point?

**Fig. 9**

Solution: Graphically, it is clear that the concavity of and changes at , so is an inflection point for and . The function
is concave up everywhere so is not an inflection point of .

If , then and . The only point at which or is undefined ( is not differentiable) is at . If , then
so is concave down. If , then so is concave up. At the concavity changes so the point is an inflection point of .

If , then and . The only point at which or is undefined
is at . If , then so is concave up. If , then so is also concave up. At the concavity **does not change** so the point
is **not an inflection point** of .

If , then and is not defined
if , but (negative number) and (positive number) so changes concavity at and is an inflection point of .

**Practice 4:** Find the inflection points of .

**Example 3:** Sketch graph of a function with , and an inflection point at . Solution: Two solutions are given in Fig. 10.

**Fig. 10**