C2: Increasing and
Decreasing Functions
Learning Objective: to use
differentiation to identify the nature
of a function
Starter:
Revise quadratics:
Solve
1. x2 + 5x + 6 = 0
2. 2x2 - 5x -3 = 0
3. x2 + 4x -5 > 0
4. x2 + 7x + 12 < 0
The gradient of a curve
The gradient of a curve at a point is given by
the gradient of the tangent at that point.
Look at how the gradient changes as we move along a curve:
Increasing Functions:
An increasing function is when
the gradient is positive at all
points on the graph.
Example : f(x) = x3 + 24x + 3
f’(x) = 3x2 + 24
As x2 ≥ 0 for all real x, 3x2 + 24 ≥ 0
So f(x) is an increasing function.
Decreasing Functions:
A decreasing function is when
the gradient is negative at all
points on the graph.
Example : f(x) = x3 + 3x2 – 9x
f’(x) = 3x2 + 6x – 9
= 3(x2 + 2x – 3)
= 3(x+ 3)(x – 1)
f’(x) < 0 when (x + 3)( x – 1) < 0
So f(x) is a decreasing function when -3 < x < 1
This is the graph of a function which is increasing for x < 0,
and is decreasing for x > 0.
At x = 0 the gradient is zero and the function is said to be
stationary.A decreasing function is when the gradient is
negative at all points on the graph.
Task 1: Find the values of x for which f(x) is an
increasing function
1.
2.
3.
4.
5.
3x2 + 8x + 2
4x – 3x2
5 – 8x – 2x2
3 + 3x – 3x2 + x3
x4 + 2x2
Find the values of x for which f(x) is a decreasing function
1. x2 – 9x
2. 2x3 – 3x2 -12x
3. x + 25/x