# Second Derivatives and Beyond

### Topics

## Introduction to Second Derivatives And Beyond - At A Glance:

A **point of inflection** or **inflection point**, abbreviated IP, is an *x*-value at which the concavity of the function changes. In other words, an IP is an *x*-value where the sign of the second derivative changes. It might also be how we'd describe Peter Brady's voice.

The usual way to look for inflection points of *f* is to

- find
*f*"

- find all
*x*-values where*f*" is zero or undefined, and

- check each such
*x*-value to see if the sign of*f*" changes there.

Again, we can use graphs to check our work. An inflection point where the function goes from concave up to concave down looks something like this:

An inflection point where the function goes from concave down to concave up looks something like this:

While any point at which *f *' is zero or undefined is a critical point, a point at which *f *" is zero or undefined is *not* necessarily an inflection point. You can think of the points where *f *" is zero or undefined as possible inflection points, but then you need to check each possible inflection point to see if it's a real inflection point.

**Be Careful:** Because *f *"(*c*) = 0 or is undefined doesn't mean *c* is an inflection point. *f *" must have different signs to either side of *c*.

There are two main ways to figure out what the sign of *f *" is doing to either side of a possible inflection point *c*.

#### Example 1

Find all points of inflection for the function |

#### Example 2

Find all inflection points for the function |

#### Example 3

Suppose we started with a function
What are the inflection points of |

#### Exercise 1

For the function, find all points of inflection or determine that no such points exist.

*f* (*x*) = sin *x*

#### Exercise 2

For the function, find all points of inflection or determine that no such points exist.

*f* (*x*) = *e*^{x}

#### Exercise 3

For the function, find all points of inflection or determine that no such points exist.

#### Exercise 4

For the function, find all points of inflection or determine that no such points exist.

#### Exercise 5

For the function, find all points of inflection or determine that no such points exist.

*f* (*x*) = 5*x* + 2

#### Exercise 6

For the function, find all points of inflection or determine that no such points exist.

*f* (*x*) = *x*^{1/3}

#### Exercise 7

For the function, find all points of inflection or determine that no such points exist.

*f* (*x*) = *xe*^{x}

#### Exercise 8

For the function, find all points of inflection or determine that no such points exist.

#### Exercise 9

For the function, find all points of inflection or determine that no such points exist.

*f* (*x*) = *x ln* *x* for *x* > > 0

#### Exercise 10

For the function, find all points of inflection or determine that no such points exist.

The logistic equation