4.6Minimum and Maximum Values of Functions

Greg Kelly, Hanford High School, Richland, Washington

Borax Mine, Boron, CAPhoto by Vickie Kelly, 2004

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

2y x

,D Absolute Minimum

No AbsoluteMaximum

2y x

0,2D Absolute Minimum

AbsoluteMaximum

2y x

0,2D No Minimum

AbsoluteMaximum

2y x

0,2D No Minimum

NoMaximum

Extreme Value Theorem:

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

Maximum & minimumat interior points

Maximum & minimumat endpoints

Maximum at interior point, minimum at endpoint

Finding Maximums and Minimums Analytically:

1 Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points.

2 Find the value of the function at each critical point.

3 For closed intervals, check the end points as well.For open intervals, check the limits as x approaches the end points to determine if min or max exist within interval.

EXAMPLE FINDING ABSOLUTE EXTREMA

Find the absolute maximum and minimum values of on the interval . 2/3f x x 2,3

2/3f x x

132

3f x x

3

23

f xx

There are no values of x that will makethe 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 aclosed interval, we also must check theendpoints.

2/ 3f x x 2,3D

At: 2x 232 2 1.5874f

At: 3x 233 3 2.08008f

2/ 3f x x 2,3D

At: 2x 232 2 1.5874f

At: 3x 233 3 2.08008f

0 0f At: 0x

Absoluteminimum:

Absolutemaximum:

0,0

3,2.08

EXAMPLE FINDING ABSOLUTE EXTREMA

Find the absolute maximum and minimum values of on interval . 2 3 1f x x x ,

2 3 1f x x x

2 3f x x Critical point at x = 1.5.

Because the function is defined over aopen interval, we also must check thelimits at the endpoints.

21.5 1.5 3 1.5 1 3.25f

lim ( )x

f x

lim ( )x

f x

Absoluteminimum:

Absolutemaximum:

1.5, 3.25

None

21.5 1.5 3 1.5 1 3.25f

lim ( )x

f x

lim ( )x

f x