Extreme Value Theorem

Chapter 5:
Application of Derivative

5.1: Extreme Values of functions

​What you will learn..
Absolute (Global)Extreme Values.
Local (Relative) Extreme Values.
Finding Extreme Values.

​Example 1:
Finding the extreme value

If f is continuous on a closed interval [a,b], then f has both maximum and minimum values on the interval.

​Theorem 1:
The Extreme Value Theorem

​​Exercises :
Find the extreme value:

​​Exercises :
Find the extreme value:

​​Exercises :
Find the extreme value:

Multiple Choice

Identify the extrema.

• Relative max (1, 4), Relative Min (3, 0)

• Relative max (1, 4), Absolute Min (3, 0)

• Absolute max (1, 4), Relative Min (3, 0)

• Relative max (0, 0), Relative Min (3, 0)

Multiple Choice

What type of extrema is (-3,2). Be as precise as possible.

• Absolute minimum

• Absolute maximum

• relative minimum

• Relative maximum

Multiple Choice

Where do you find relative minimums and relative maximums?

• zeros
• y-intercepts
• turning points
• by degree

Multiple Choice

What is the relative maximum value?

• y = 4
• y = 2
• y = 1
• y = -1

Multiple Choice

How many extrema (maxes and mins) are in the picture?

• 2
• 3
• 4
• 5

Multiple Choice

There is a relative maximum at:

• (-1,3)

• (3,1)

• (-2.5,-4)

• (1.5,-4)

​Finding Extreme values

​​ Finding Extreme Values

Multiple Choice

Find the critical points of f(x) = 2x⁴- 4x² + 1

• x= 0
• x = -1, 1
• x = -1, 0, 1
• no critical points

Multiple Choice

Find the extreme values of the function and where they occur.

$$y=2x^2-8x+9$$

• Max. value 2 at x=1

• Min. value 1 at x=2

• None

• Local max at (0,-1)

Multiple Choice

Find the extreme values of the function.

$$y=\sqrt[]{x^2-1}$$

• Local max at (0,-1)

• Min value 0 at x=-1,3

• Min value 0 at x=1,-1

• Local min at (0,1)

​ Example 5:

Multiple Choice

Find the critical point and determine the local extreme values

• critical point x=1

  derivative (undefined)

  minimum y=2

• critical point x=-2

  derivative (undefined)

  maximum y=4