The document discusses increasing and decreasing functions. It defines an increasing function as one where the gradient (dy/dx) is always positive, and a decreasing function as one where the gradient is always negative. To determine if a function is increasing or decreasing, you find where the derivative (dy/dx) is equal to 0 to find critical points, and use a nature table to analyze the sign of the derivative on each interval. The derivative also needs to be analyzed to determine if a function is always increasing or decreasing.

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. increasing
increasing
decreasing
need to find SPs
dy
/dx = 0
Increasing → dy
/dx is +ve
Decreasing → dy
/dx is -ve

4. Ex y = 4x3
– 3x2
+ 10
Function Decreasing?
For SPs dy
/dx = 0
dy
/dx = 12x2
– 6x
12x2
– 6x = 0
6x(2x – 1) = 0
6x = 0 or 2x – 1 = 0
x = 0 or x = ½

5. Nature Table
y = 4x3
– 3x2
+ 10
dy
/dx = 12x2
– 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 =x3
+ 7x
dy
/dx = 3x2
+ 7
Increasing as dy
/dx > 0 for all x.

7. Function always decreasing
• dy
/dx always -ve (i.e < 0)
Ex y = -6x -x3
dy
/dx = -6 - 3x2
Decreasing as dy
/dx < 0 for all x.

8. Less obvious
y = 1
/3x3
+ 3x2
+ 11x
dy
/dx= x2
+ 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
/3x3
+ 4x2
+ 17x
dy
/dx= x2
+ 8x + 17
completing square (x + 4)2
– 16 + 17
(x + 4)2
+ 1
Increasing as dy
/dx > 0 for all x.