Download 3.1 - El Camino College

Sec 3.1 Increasing and Decreasing Functions
Objective: Find the intervals in which a function is increasing or decreasing
Definition of Increasing and Decreasing Functions
Definition of increasing function
f is increasing on an interval I if fx 1  < fx 2  whenever x 1 < x 2 and x 1 and x 2 are in I.
Definition of decreasing function
f is decreasing on an interval I if fx 1  > fx 2  whenever x 1 < x 2 and x 1 and x 2 are in I.
Definition of a Critical Number
′
A critical number of a function f is a number c in the domain of f such that either f c = 0
′
or f c does not exist.
Theorem: If f has a local extremum at c, then c is a critical number of f. .
Guidelines: For finding increasing or decreasing intervals
1. Find derivative of f
2. Find all values of x where f′x = 0 or f′x is undefined
These are the c , the critical numbers of f.
3. Test sign of f′x in the regions between critical numbers
4. f′x > 0 then fxis increasing in that region interval I
f′x < 0 then fxis decreasing in that region interval I
———————————————————————————————.Example 3
a.Find the critical numbers. of the function
b. Find the open intervals on which the function is increasing or decreasing
fx = 2x 3 − 9x 2
—————————————
a. Solution
d fx = 6x 2 − 18x
step 1
dx
step 2
Set derivative =0
6x 2 − 18x = 0
6xx − 3 = 0, :
Solution is: 0, 3
the critical numbers
b. Solution
1
Make a table of the regions: intervals between the critical numbers
3, ∞
Test Intervals
−∞, 0
0, 3
Test number
-1
1/2
4
+
-
+
′
Sign of f x = 6xx − 3
Conclusion fx
f0 = 0
increasing decreasing increasing
f3 = −27
y
-3
-2
-1
1
2
3
4
5
x
-50
-100
-150
———————————————————-:
Do NOW check point 3 (altered)
Find the critical numbers and the intervals only
fx = x 3 − x
————————
——————————————–
.Example 4
Find the open intervals on which the function is increasing or decreasing
fx = x 3 − 32 x 2
—————————————————
Solution:
y = fx = x 3 − 32 x 2
step1y′ = 3x 2 − 3x
step2; Set derivative 3x 2 − 3x =0
′
y′ = f x = 3xx − 1 = 0, critical numbers: 0 and 1
step3: make table
Test Intervals
−∞, 0
0, 1
1, ∞
Test number
-1
1/2
2
-
+
′
f x = 3xx − 1 +
Conclusion f(x)
fx = x −
3
3
2
x
increasing decreasing increasing
2
2
y
2
-1
1
2
x
-2
Do NOW cp4
fx = x 3 − 12x
————————
——————————————–
.Example 5
Find the open intervals on which the function is increasing or decreasing
2
fx = x 2 − 4 3
—————————————————
Solution:
2
y = fx = x 2 − 4 3
dy
step1: dx = 43 3 x2
x −4
step2: Set derivative =0
′
f x = 43 3 x2
= 0 critical number: 0
x −4
′
step 3 f x dosen’t exist at x=2,-2
critical numbers are -2,0,2
Test Intervals
−∞, −2
−2, 0
0, 2
2, ∞
Test number
-3
-1
1
3
-
+
-
+
′
f x =
4
3
x
3
x 2 −4
Conclusion f(x) decreasing increasing decreasing increasing
y6
4
2
-4
-2
0
2
4
x
2
fx = x 2 − 4 3
——————————–
Do Now checkpoint 5
2
fx = x 3
see your table
———————————————–
3
Example 7 Test an Increasing Function
Show that fx = x 3 − 3x 2 + 3x
in increasing on the entire real line
—————————————————
Solution:
y = fx = x 3 − 3x 2 + 3x
dy
= 3x 2 − 6x + 3
step1:
dx
step 2:Set derivative =0
′
f x
3x 2 − 6x + 3 = 0
3x 2 − 2x + 1 = 0
3x − 1 2 = 0
critical number x=1
Test Intervals
−∞, 1
1, ∞
Test number
0
2
f x = 3x − 1 2 +
+
′
Conclusion f(x)
y
increasing increasing
4
2
-1
1
2
3
x
-2
———————————————–
Do NOW checkpoint 7 (if time)
Hint- Homework: Sec 3.1 no. 27,
fx = x 2 − 1
y
4
2
-4
-2
-2
2
4
x
-4
Note: In the inteval (-1,0) and (0,-1),where x 2 − 1 > 0
(never a negative under the
sign)
y’ is undefined
Consider only the intervals −∞, −1 1, ∞
4