Critical Numbers and Finding Extrema. Critical Numbers Example 1: Example 2: 1.Take the derivative...
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Critical Numbers and Finding Extrema
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Transcript of Critical Numbers and Finding Extrema. Critical Numbers Example 1: Example 2: 1.Take the derivative...
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
