APPLICATION OF
DIFFERENTIATION
INCREASING AND DECREASING FUNCTION
MINIMUM & MAXIMUM VALUES
Increasing & Decreasing
function
2 ND D I F F E R E N T I A T I O N
Determine set values of x in which the function is increasing and
y
decreasing
40
20
x
-6
-4
-2
2
-20
-40
-60
The function decreases when
The function increases when
-80
4
The nature of stationary point
2 ND D I F F E R E N T I A T I O N
10
y
Find the point on the curve when8 its
tangent line has a gradient of 0. 6
4
2
x
-10
-8
-6
-4
-2
2
-2
-4
-6
-8
-10
Stationary point is a point where its
tangent line is either horizontal or
vertical.
How is this related to 2nd differentiation?
4
6
8
10
10
y
8
6
4
2
x
-10
-8
-6
-4
-2
2
-2
Find the point on the curve when its
tangent line has a gradient of 0.
-4
-6
-8
-10
How is this related to 2nd differentiation?
4
6
8
10
Find the point on the curve
when its
y
5.5
tangent line has a gradient
of 0.
5
4.5
4
3.5
3
2.5
2
1.5
x
-2
-1.5
-1
-0.5
1
0.5
1
1.5
2
What is the nature of this point?
This point is neither maximum nor
minimum point and its called
STATIONARY POINT OF INFLEXION
How do we apply these concepts?
Find the coordinates of the stationary points on the curve
y = x3  3x + 2 and determine the nature of these points.
Hence, sketch the graph of y = x3  3x + 2 and determine the set
values of x in which the function increases and decreases.
What are the strategies to solve this question?
5
y
4
3
2
1
–6
–4
–2
2
–1
–2
–3
How do we apply these concepts to solve real-life
problems?
An open tank with a square base is to be made from a thin
sheet of metal. Find the length of the square base and the
height of the tank so that the least amount of metal is used to
make a tank with a capacity of 8 m3.
What are the strategies to solve
this question?
• Derive a function from surface area and/ or
volume area.
• Express the function in one single term (x)
• Use the function to identify maximum or
minimum value.
h
x
x
An open tank with a square base is to be made from a
thin sheet of metal. Find the length of the square base
and the height of the tank so that the least amount of
metal is used to make a tank with a capacity of 8 m3.
The Volume shows relationship between
the height (h) and length (x) of the tank
Express S in terms of x
h
x
x
Since the amount of the metal needed
depends on the surface area of the
tank, the area of metal needed is