1. Application of Differentiation
Determination of
Maximum
&
Minimum Point

2. Objective After completing this topic students should be able to:
Determine extreme points using “first derivative” f’(x).
Higher derivative of a given function (e.g. f”(x)).
Determine the nature of a given point using “first derivative” f’(x) and “second derivative” f”(x).

3. Maximum & Minimum Values
(1, 7)
(-1, -2) A B

4. Find any maximum and/or minimum points for the graph of:
Without seeing the graph, how could we tell which of these two points is the maximum and which is the minimum?????
In order to locate these points precisely, we need to find the values of x for which

5. f (x) f ´(x) > 0 f ´(x) < 0 f ´(x) > 0 increasing decreasing
Remember this graph from when we first discussed derivatives?
maximum
minimum
f (x)
f ´(x) > 0
f ´(x) < 0
f ´(x) > 0
increasing
decreasing
increasing
f ´(x)

6.  + + max min Find any maximum and/or minimum points for the graph of:
So let’s test the intervals:
Important note: The above number line without an explanation will not be considered sufficient justification on the exam

7. Find any maximum and/or minimum points for the graph of:

8. Also called a global maximum Also called a relative maximum
Note:
Absolute maximum
Local maximum
Also called a global maximum
Also called a relative maximum
Note that an absolute max/min is already a local max/min
Local minimum
Also called a relative minimum

9. Increasing & Decreasing Functions, Stationary Points (page 375 Text book)
A function is increasing when
f ′(x) > 0
A function is decreasing when
f ′(x) < 0
Stationary points occur when
f ′(x) = 0

10. Stationary Points
The nature of a stationary point depends on the gradient on either side of it. Stationary points have f ′ (x) = 0 i.e. the gradient is 0 so the gradient function is horizontal at that point.

11. Maximum Turning Point x  TP f ′ (x) Graph T P + ve − ve Max TP
A maximum turning point (max TP) occurs when the gradient either side of a stationary point changes from positive to negative. x  TP
f ′ (x)
Graph
T P
+ ve
− ve
Max TP

12. Minimum Turning Point x  TP f ′ (x) Graph − ve + ve T P Min TP
A minimum turning point (min TP) occurs when the gradient either side of a stationary point changes from negative to positive. x  TP
f ′ (x)
Graph
− ve
+ ve
T P
Min TP

13. Rising Horizontal Point of Inflexion
Text book page 375 : Turning Point
A rising point of inflexion (P I) occurs when the gradient either side of a stationary point remains positive. x  TP
f ′ (x)
Graph
+ ve
+ ve
P I
Rising P I

14. Falling Horizontal Point of Inflexion
A falling point of inflexion (P I) occurs when the gradient either side of a stationary point remains negative. x  TP
f ′ (x)
Graph
− ve
- ve
P I
Falling P I

15. f (x) f ´(x) > 0 f ´(x) < 0 f ´(x) > 0 increasing decreasing
What is the graph of gradient function?
(how does it look like?)
maximum
minimum
f (x)
f ´(x) > 0
f ´(x) < 0
f ´(x) > 0
increasing
decreasing
increasing
f ´(x)

16. Example f is increasing when the derivative is positive.
Use the graph of f '(x) below to determine when f is increasing and decreasing.
f is increasing when the derivative is positive.
f ' (x)
f is increasing when the derivative is positive. x
f is decreasing when the derivative is negative.

17. The graph of f is shown below. Sketch a graph of the derivative of f
The graph of f is shown below. Sketch a graph of the derivative of f. (see examples)

18. The Nature of Stationary Points (conclusion)
Solve the equation f ′ (x) = 0 and
consider the gradient f ′ (x) around each stationary point:
If f ′ (x) is positive then the graph is increasing
If f ′ (x) is negative then the graph is decreasing

19. Find the sign of the derivative on each interval.
Example
Find where the function is increasing and where it is decreasing.
Find the derivative.
Find the sign of the derivative on each interval. -1 2

20. Higher-Order Derivatives
The derivative f ′ of a function f is also a function.
As such, f ′ may also be differentiated.
Thus, the function f ′ has a derivative f ″ at a point x.
The function f ″ obtained in this manner is called the second derivative of the function f, just as the derivative f ′ of f is often called the first derivative of f.
By the same method, you may consider the third, fourth, fifth, etc. derivatives of a function f.

21. Higher-Order Derivatives
Examples:
Find the second derivative of the function
f(x) = (2x2 +3)3/2
Solution
Using the general power rule (chain rule) we get the first derivative:

22. Higher-Order Derivatives
Examples:
Find the second derivative of the function f(x) = (2x2 +3)3/2
Solution
Using the product rule we get the second derivative:

23. OPTIMIZATION
The second derivative test can be useful for determining whether a given stationary point is a local maximum or a local minimum.
If f’’(x)< 0 at a stationary point then the point is a local maximum point.
If f’’(x)>0 at a stationary point then the point is a local minimum point.

24. Turning Points and Points of Inflection
(Text book : page 379)
Stationary points of a cubic curve
There are three types of stationary point: maxima, minima and points of inflexion. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x
Stationary Point f’(x)= 0
Local Maximum
Decreasing f’(x)< 0
Increasing f’(x)> 0
Pt. of Inflection
Increasing f’(x)> 0
Local Minimum
Stationary Point f’(x)= 0
The second derivative test can be useful for determining whether a given stationary point is a local maximum or a local minimum.
If f’’(x)< 0 at a stationary point then the point is a local maximum point.
If f’’(x)>0 at a stationary point then the point is a local minimum point.
If f’’(x)= 0 then the point is point of inflexion.

25. Classifying Turning Points: The Second Derivative Test
***
Classifying Turning Points: The Second Derivative Test -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x
At a max/min point f’(x)=0
At a max point f”(x) < 0
At a min point f”(x) > 0 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 x

26. Optimisation
Differentiation can be used to solve problems which require maximum or minimum values.
Problems typically cover topics such as areas, volumes and rates of change. They often involve having to establish a suitable formula in one variable and then differentiating to find a maximum or minimum value. This is known as Optimisation.

27. Maximum and Minimum Problems
Optimisation
Maximum and Minimum Problems
Calculus is an important tool in solving maximum–minimum problems.
From a 30 cm × 30 cm sheet of cardboard, square corners are cut out so that the sides are folded up to make a box.
What is the maximum volume of the box?

28. Optimisation Example

29. Optimisation Example

30. Optimization Optimization falls into two categories:
Maximization (ie: production, profits, utility, happiness, grades, health, employment, etc.)
Minimization (ie: costs, pollution, disutility, unemployment, sickness, homework, etc.) 6

31. Optimizing in 3 Steps There are three steps for optimization:
FIRST ORDER CONDITION (FOC) Find where f’(x)=0. These are potential maxima/minima.
SECOND ORDER CONDITION (SOC) Evaluate f’’(x) at your potential maxima/minima. This determines if (1)’s solutions are maxima/minima/inflection points
Co-Ordinates Obtain the co-ordinates of your maxima/minima 6