Increasing decreasing functions

Increasing/Decreasing Functions

1. Block 1
Increasing/Decreasing
Functions

2. What is to be learned?
•What is meant by increasing/decreasing
functions
•How we work out when function is
increasing/decreasing
•How to show if a function is always
increasing/decreasing

3. in
c
re
a
s
in
g
i n
c
r e
a
s
i n
g
d
e
c
r
e
a
s
i
n
g
need to find SPs
dy
/
dx
= 0
Increasing →
dy
/
dx
is +ve
Decreasing →
dy
/
dx
is -ve

4. Ex y = 4x
3
– 3x
2
+ 10
FunctionDecreasing?
For SPs
dy
/
dx
= 0
dy
/
dx
= 12x
2
– 6x
12x
2
– 6x = 0
6x(2x – 1) = 0
6x = 0 or 2x – 1 = 0
x = 0 or x = ½

5. Nature Table
y = 4x
3
– 3x
2
+ 10
dy
/
dx
= 12x
2
– 6x
= 6x(2x – 1)

SPs at x = 0 and ½
x
0
dydy
//
dxdx = 6x(2x – 1) = 6x(2x – 1)
0
-1 ¼
= + = -
Slope
Max TP
at x = 0
½
1
= +
0
Min TP
at x = ½
- X - + X - + X +
Decreasing 0 < x < ½

6. Function always increasing?
•
dy
/
dx
always +ve (i.e > 0)
Ex y =

x
3
+ 7x

dy
/
dx
= 3x
2
+ 7
Increasing as
dy
/
dx
> 0 for all x.

7. Function always decreasing
•
dy
/
dx
always -ve (i.e < 0)
Ex y =

-6x -

x
3

dy
/
dx
= -6 - 3x
2
Decreasing as
dy
/
dx
< 0 for all x.

8. Less obvious
y =
1
/
3
x
3
+ 3x
2
+ 11x

dy
/
dx
= x
2
+ 6x + 11
completing square (x + 3)
2
– 9 + 11
(x + 3)
2
+ 2
Increasing as
dy
/
dx
> 0 for all x.

9. Increasing/Decreasing Functions
•Increasing → Gradient +ve (
dy
/
dx
> 0)
•Decreasing → Gradient -ve (
dy
/
dx
< 0)
•Find SPs (only need x values)
•Completing square can be handy tactic

10. Ex
y =
1
/
3
x
3
+ 4x
2
+ 17x

dy
/
dx
= x
2
+ 8x + 17
completing square (x + 4)
2
– 16 + 17
(x + 4)
2
+ 1
Increasing as
dy
/
dx
> 0 for all x.