- Topics At a Glance
- Second Derivatives via Formulas
- Third Derivatives and Beyond
- Concavity
- Concave Up
- Concave Down
- No Concavity
- Special Points
- Critical Points
- Points of Inflection
**Extreme Points and How to Find Them**- Finding & Classifying Extreme Points
- First Derivative Test
- Second Derivative Test
- Local vs. Global Points
- Using Derivatives to Draw Graphs
- Finding Points
- Finding Shapes
- Connecting the Dots
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

The maximum value of the function *f* (*x*) = -*x*^{2 }– 1 is *y* = -1:

The maximum value of the function *f* (*x*) = cos *x* is *y* = 1:

**Extreme points**, also called **extrema**, are places where a function takes on an **extreme value**—that is, a value that is especially small or especially large in comparison to other nearby values of the function. Extrema look like the tops of hills and the bottoms of valleys. Time to go hiking.

There are two types of extreme points, **minima** (the valleys) and **maxima** (the hills).

Extreme points can be **local** or **global**, but we'll talk about this later.

We need to define minimum and maximum values without the *on an interval* bit.

A **minimum value** of a function is a *y*-value of the function that is as low, or lower, than other values of the function nearby. A minimum looks like a valley:

The plural of minimum is **minima**.

The minimum value of the function *f* (*x*) = *x*^{2} + 1 is *y* = 1:

The minimum value of the function *f* (*x*) = cos *x* is *y* = -1:

A function may have multiple minima.

The function graphed below has two minima: *y* = 0 and *y* = 1.

A function may have infinitely many minima.

The function graphed below has infinitely many minima:

A function may have no minima at all.

The function *f* (*x*) = -*x*^{2} has no minima, because for every value of the function there are smaller values nearby:

**Be Careful:**There is a difference between a minimum of a function (a *y*-value) and where that minimum occurs (an *x*-value).

The minimum value of the function *f* (*x*) = *x*^{2} + 1 is *y* = 1, and this minimum occurs at *x* = 0:

The function *f* (*x*) = cos *x* has only one minimum value, *y* = -1. However, this minimum value occurs at infinitely many places, as it occurs at *x* = π + 2*n*π for every integer *n*:

A function may have multiple maxima.

The function graphed below has two maxima: *y* = 2 and *y* = 3.

A function may have infinitely many maxima.

The function graphed below has infinitely many maxima:

A function may have no maxima at all.

The function *f* (*x*) = *x*^{2} has no maxima, because for every value of the function there are larger values nearby:

**Be Careful:**There is a difference between a maximum of a function (a *y*-value) and where that maximum occurs (an *x*-value).

The maximum value of the function *f* (*x*) = -*x*^{2 }– 1 is *y* = -1, and this maximum occurs at *x* = 0:

The function *f* (*x*) = cos *x* has only one maximum value, *y* = 1. However, this maximum value occurs at infinitely many places, as it occurs at *x* = 2nπ for every integer *n*:

Exercise 1

For the function, (a) find all minima of the function, if any, and (b) determine the *x*-value(s) at which each minima occurs.

Exercise 2

For the function, (a) find all minima of the function, if any, and (b) determine the *x*-value(s) at which each minima occurs.

Exercise 3

For the function, (a) find all minima of the function, if any, and (b) determine the *x*-value(s) at which each minima occurs.

Exercise 4

*x*-value(s) at which each minima occurs.

Exercise 5

*x*-value(s) at which each minima occurs.

Exercise 6

*x*-value(s) at which each minima occurs.

*f* (*x*) = 2 - *x*^{2}

Exercise 7

*x*-value(s) at which each minima occurs.

*f* (*x*) = (*x* - 4)^{2} + 3

Exercise 8

*x*-value(s) at which each minima occurs.

*f* (*x*) = 5

Exercise 9

*x*-value(s) at which each minima occurs.

*f* (*x*) = 3*x* + 7

Exercise 10

*x*-value(s) at which each minima occurs.

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

Exercise 11

These questions deal with characteristics of the first and second derivatives of a function at and near a minimum value.

a. Below is a graph of a function *f* with a minimum at *x* = x_{0}. Determine the sign of the derivative *f*' at each labelled *x*-value.

b. Assume *f* is defined and twice differentiable on the whole real line. Around a minimum of the function *f*, is *f* concave up or concave down?

Exercise 12

These questions deal with characteristics of the first and second derivatives of a function at and near a maximum value.

a. Below is a graph of a function *f* with a maximum at *x* = *x*_{0}. Determine the sign of the derivative *f*' at each labelled *x*-value.

b. Assume *f* is defined and twice differentiable on the whole real line. Around a maximum of the function *f*, is *f* concave up or concave down?