4.1

Extreme Values of Functions

Greg Kelly, Hanford High School, Richland, Washington

Borax Mine, Boron, CA

Photo by Greg Kelly, 2019

4.1

Extreme Values of Functions

Greg Kelly, Hanford High School, Richland, Washington

Borax Plant, Boron, CA

Photo by Vickie Kelly, 2019

The textbook gives the following example at the start of chapter 4:

Of course, this problem isn’t entirely realistic, since it is unlikely that you would have an equation like this for your car.

We could solve the problem graphically:

We could solve the problem graphically:

On the TI-89, we use F5 (math), 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

We could solve the problem graphically:

On the TI-89, we use F5 (math), 4: Maximum, choose lower and upper bounds, and the calculator finds our answer.

Notice that at the top of the curve, the horizontal tangent has a slope of zero.

Traditionally, this fact has been used both as an aid to graphing by hand and as a method to find maximum (and minimum) values of functions.

Absolute extreme values are either maximum or minimum points on a curve.

Even though the graphing calculator and the computer have eliminated the need to routinely use calculus to graph by hand and to find maximum and minimum values of functions, we still study the methods to increase our understanding of functions and the mathematics involved.

They are sometimes called global extremes.

They are also sometimes called absolute extrema.

(Extrema is the plural of the Latin extremum.)

Extreme values can be in the interior or the end points of a function.

Absolute Minimum

No Absolute

Maximum

Absolute Minimum

Absolute

Maximum

No Minimum

Absolute

Maximum

No Minimum

No

Maximum

Extreme Value Theorem:

If f is continuous over a closed interval, then f has a maximum and minimum value over that interval.

Maximum & minimum

at interior points

Maximum & minimum

at endpoints

Maximum at interior point, minimum at endpoint

Local Extreme Values:

A local maximum is the maximum value within some open interval.

A local minimum is the minimum value within some open interval.

Absolute minimum

(also local minimum)

Local maximum

Local minimum

Absolute maximum

(also local maximum)

Local minimum

Local maximum

Local minimum

Absolute maximum

(also local maximum)

Local Extreme Values:

If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if exists at c, then

Critical Point:

A point in the domain of a function f at which

or does not exist is a critical point of f .

Note:

Maximum and minimum points in the interior of a function always occur at critical points, but critical points are not always maximum or minimum values.

There are no values of x that will make

the first derivative equal to zero.

The first derivative is undefined at x=0,

so (0,0) is a critical point.

Because the function is defined over a

closed interval, we also must check the

endpoints.

To determine if this critical point is

actually a maximum or minimum, we

try points on either side, without

passing other critical points.

Since 0<1, this must be at least a local minimum, and possibly a global minimum.

To determine if this critical point is

actually a maximum or minimum, we

try points on either side, without

passing other critical points.

Since 0<1, this must be at least a local minimum, and possibly a global minimum.

Absolute minimum (0,0)

Absolute maximum (3,2.08)

Finding Maximums and Minimums Analytically:

Critical points are not always extremes!

(not an extreme)

(not an extreme)

π