extreme values of functions
DESCRIPTION
Extreme Values of Functions. Section 4.1b. Do Now. Find the extreme values of:. First, check the graph What does it suggest?. What is the domain of the function?. Since there are no endpoints, all extreme values must occur at critical points. Find the derivative:. Do Now. - PowerPoint PPT PresentationTRANSCRIPT
Section 4.1b
EXTREME VALUES OF FUNCTIONS
Find the extreme values of:
2
1
4f x
x
1 224 x
Do Now
First, check the graph What does it suggest?
What is the domain of the function? : 2, 2D Since there are no endpoints, all extreme values must occurat critical points. Find the derivative:
3 221 4 22
f x x x
3 224
x
x
Find the extreme values of:
2
1
4f x
x
1 224 x
Do Now
No Maxima, Minimum of 1/2 at x = 0
3 224
xf xx
The only critical point in thedomain is at x = 0…
Check the original function:
2
104 0
f
12
As x moves away from 0 on eitherside denominator gets smallerValue of f increases the graphrises We have a minimum!!!
Extrema can occur at critical points and endpoints, butnot every critical point or endpoint automaticallysignals an extrema!!!
An Important Note
3 5( )k x x 2 3x
Ex: Find the extrema of the following function using bothanalytic and graphical methods:
2 53( )5
k x x
Start with the graph…
2 5
35x
Derivative is never zero,and is undefined at x = 0.
Extrema can occur at critical points and endpoints, butnot every critical point or endpoint automaticallysignals an extrema!!!
An Important Note
3 5( )k x x 2 3x
Ex: Find the extrema of the following function using bothanalytic and graphical methods:
Start with the graph…
Critical Point: 0x Endpoint: 3x But only the endpoint signals an extrema:
Maximum of at3 53 3x
More Practice Problems
3 2 4f x x x
36 6 62 43 3 3
f
Find the extreme values of the given function.
23 2f x x
Critical points:23
x Consider the graph! 6
3
2 6 2 6 49 3
4 6 369
5.089
More Practice Problems
3 2 4f x x x
36 6 62 43 3 3
f
Find the extreme values of the given function.
23 2f x x
Critical points:23
x Consider the graph! 6
3
2 6 2 6 49 3
4 6 369
2.911
More Practice Problems
3 2 4f x x x
Local Max of at
Find the extreme values of the given function.
23 2f x x
4 6 369 6
3x
Local Min of at4 6 369
63
x
More Practice ProblemsFind the extreme values of the given function.
25 2 ,
2,x
f xx
11
xx
First, sketch the graph…
It appears that the derivative is zero at x = 0, and doesnot exist at x = 1. There appears to be a local max of5 at x = 0 and a local min of 3 at x = 1.
More Practice ProblemsFind the extreme values of the given function.
25 2 ,
2,x
f xx
11
xx
Critical Points atx = 0 and x = 1
Local Max of 5 at x = 0,Local Min of 3 at x = 1
How do we confirm this result analytically?
4 , 11, 1x x
f xx
Derivative is zero at x = 0Left- and right-hand derivativesare not equal at x = 1
More Practice ProblemsFind the extreme values of the given function.
2 3y x x Start by checking the graph…
2 1 1 2 32 3
dy x x xdx x
2 4 32 3 2 3
x xxx x
25 122 3x x
x
Derivative is zero at x = 0 and x = 12/5
Derivative is undefined at x = 3
More Practice ProblemsFind the extreme values of the given function.
2 3y x x Start by checking the graph…25 12
2 3dy x xdx x
C.P. Derivative Extremum Value
x = 0x = 12/5x = 3
00
Und.
Min.Local Max.
Min.
0
04.462