SECTION 3-1
Extrema
Maximum and Minimums
Critical Values
• Where the derivative is zero or the
function does not exist.
Extrema: Let f be defined on the interval
(a,b) containing the point c.
1. f(x) has a maximum at x = c if f(c) ≥ f(x) for all x in (a,b)
• Changes from increasing to decreasing
• Horizontal tangent
Extrema: Let f be defined on the interval
(a,b) containing the point c.
2. f(x) has a minimum at x = c if f(c) ≤ f(x) for all x in (a,b)
•
•
Changes from decreasing to increasing
Horizontal tangent
Extreme Value Theorem
• If f(x) is continuous on a closed interval, then f
has both a minimum and maximum (may be the
endpoints)
Max (-2,4)
Min (7, -0.5)
Two types
1.
Relative: the curve has a horizontal tangent line at the
point, but it is not the highest or lowest point (local)
2.
Absolute: either a horizontal tangent or an endpoint w
here the curve reaches its highest or lowest value
(Global)
Determine from the graph whether f has relative
minimums, relative maximums, absolute
minimums, or absolute maximums on the closed
interval
Absolute max
Relative max
Relative max
Relative min
Absolute min
1) Show that
x2
f ( x)  2
x 4
has a minimum at (0,0).
Find the value of the derivative at the min.

x
f ' ( x) 
2

x
 4 2 x   x 2 2 x 
2
4

2
4 ( 0 )  0( 2 )
f ' ( 0) 
42
f ' ( 0)  0
2.) Show that f ( x)  ( x  2)
2
3
has a minimum at
(-2,0). Find the value of the derivative at the min.
2
1
f ' ( x)   x  2  3 1
3
2
1
f ' (2)   2  2  3
3
f ' (2) 
2
 und
3(0)
 f ' (2)  d .n.e.
Cusp
Derivatives and Extrema:
Critical Numbers: the value c contained in the
interval that gives an extrema such that
1.
2.
f (c)  0
f (c ) does not exist
3. Endpoints (if given from closed interval)
Derivatives and Extrema:
Critical Numbers: find the x-value of the point
Critical Points: find the x-value and the y-value of the
point (x,y)
3
2
f
(
x
)


2
x

3
x
3.) Find the critical numbers for
f ' ( x)  6 x  6 x  0
2
 6xx 1  0
 6x  0
x0
x 1  0
x 1
4.) Find the critical numbers for f ( x)  4  x 2

f ( x)  4  x
2

1

  2 x 
1
f ' ( x)  4  x 2
2
f ' ( x) 
x0
x0
1
2
x
4  x 
2
max& min
2
1
2
dne
4  x2  0
 4 x   0
2
2
4  x2  0
x2  4
x  2
5.) Find the critical numbers when
f ' ( x)  2 sin x cos x  2 cos x
2 sin x cos x  2 cos x  0
2 cos xsin x  1  0
2 cos x  0
sin x  1  0
cos x  0
sin x  1
x

2
x
3
2
3
x
2
6.) Find the absolute extrema for
2
f ( x)  x 3 on [-1,2]
plot the points
(1, 1), 0, 0 , 2, 1.587 
2  13
f ' ( x)  x
3
f ' ( x) 
2
33 x
max& min
dne
endpoints
20
33 x  0
f (1)   1 3  1
x0
f (2)  2 3  1.587
f ( 0)  0
2
2
2
3
0
∴ ab max 2,1.587 ,
ab min 0,0 By the
Extreme Value Theorem
7.) Find the absolute extrema for
1
f ( x) 
on [-3,1]
2
1 x

f ( x)  1  x

2 1

f ' ( x)  1 1  x
f ' ( x) 
 2x
2 2
 2x
1  x 
2 2
dne
endpoint
 2x  0
1  x 
x0
1 x2  0
max& min
f (0) 
1
1

1
1  02 1
2 2
x2  1
x  1
0
1
1

1  (3) 2 10
1
1
f (1) 

1  12 2
f (3) 
Homework
Page 169
# 1-10 (use instructions for 7-10)
# 11, 12, 17,19, 20, 23, 25, 33, 35,
43, 44, 45, 54, and 57-60
Worksheet 3-A