Critical Numbers and Finding Extrema. Critical Numbers Example 1: Example 2: 1.Take the derivative...
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![Page 1: Critical Numbers and Finding Extrema. Critical Numbers Example 1: Example 2: 1.Take the derivative of f(x) 2.Set the derivative equal to zero 3.Solve.](https://reader036.fdocuments.us/reader036/viewer/2022083006/56649f385503460f94c55147/html5/thumbnails/1.jpg)
Critical Numbers and Finding Extrema
![Page 2: Critical Numbers and Finding Extrema. Critical Numbers Example 1: Example 2: 1.Take the derivative of f(x) 2.Set the derivative equal to zero 3.Solve.](https://reader036.fdocuments.us/reader036/viewer/2022083006/56649f385503460f94c55147/html5/thumbnails/2.jpg)
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
![Page 3: Critical Numbers and Finding Extrema. Critical Numbers Example 1: Example 2: 1.Take the derivative of f(x) 2.Set the derivative equal to zero 3.Solve.](https://reader036.fdocuments.us/reader036/viewer/2022083006/56649f385503460f94c55147/html5/thumbnails/3.jpg)
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
![Page 4: Critical Numbers and Finding Extrema. Critical Numbers Example 1: Example 2: 1.Take the derivative of f(x) 2.Set the derivative equal to zero 3.Solve.](https://reader036.fdocuments.us/reader036/viewer/2022083006/56649f385503460f94c55147/html5/thumbnails/4.jpg)
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