Critical Numbers and Finding Extrema

Critical Numbers

04

1

14

14

140

4)(

4)(

14)(

2

2

2

2

2

1

x

x

x

x

x

xxf

xxxf

xxxf

0

60

6)(

33)( 2

x

x

xxf

xxf

Example 1:

Example 2:1. Take the derivative

of f(x)2. Set the derivative

equal to zero3. Solve for x

*Remember to test the regions to make sure the critical numbers are valid

you can’t take the square root of a negative number, so the only critical number is 0

Finding Extrema(closed interval, continuous function)

1. Find critical numbers2. Find endpoint values3. Find the function values for the x values

determined in step one and two4. Identify extrema

a. Largest overall function value is the absolute maxb. Smallest overall function value is the absolute

min

Finding Extrema Example

Find the absolute min and max for the function on the interval [-3,5] :

2,2

)2)(2(30

)4(30

123)(

112)(

2

2

3

x

xx

x

xxf

xxxf

66)5(

10)3(

15)2(

17)2(

f

f

f

f

Absolute max is 66 and occurs at x=5 and the absolute min is -15 and occurs at x=2