# Second Derivatives and Beyond

# Extreme Points and How to Find Them

### Sample Problem

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

### Sample Problem

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**.

### Sample Problem

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

### Sample Problem

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

A function may have multiple minima.

### Sample Problem

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

A function may have infinitely many minima.

### Sample Problem

The function graphed below has infinitely many minima:

A function may have no minima at all.

### Sample Problem

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).

### Sample Problem

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

### Sample Problem

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.

### Sample Problem

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

A function may have infinitely many maxima.

### Sample Problem

The function graphed below has infinitely many maxima:

A function may have no maxima at all.

### Sample Problem

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).

### Sample Problem

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

### Sample Problem

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*: