SM015 - 10.1 (Notes Solutions)

10.1 Extremum Problems

Friday, July 10, 2020 3:14 PM

Learning Outcomes:
At the end of the lesson, students should be able to
(a) Find the critical points..
(b) Find the relative extremum using the first derivative test.
(c) Find the relative extremum using the second derivative test.
(d) Solve optimization problems.

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Critical Point

 A function has a critical value at if

(i) is defined &

(ii) or is undefined.

 The point is called the critical point.

Stationary Point

 If a function is defined at and at ,

then the point is called as stationary point .

[ is defined and ]

 Graphically,

10.1 Extremum Problems Page 1

Example 1

Saturday, May 02, 2020 10:20 PM

Find the critical points of
Solution:

Note that is undefined when

The critical points when are undefined are

The critical point when is
Critical points :

10.1 Extremum Problems Page 2

Example 2

Saturday, May 02, 2020 10:20 PM

Find the stationary points for
Solution:

Stationary points :

10.1 Extremum Problems Page 3

Extremum Points

Saturday, May 02, 2020 2:40 PM

Remarks:
• Turning Point Stationary Point
• Inflection point may NOT be a
stationary point
• For a function ,
-
-

TWO methods to determine the type of stationary point:
a) First Derivative Test

b) Second Derivative Test

Steps to determine types of extremum points
a) Find the derivative of the function, i.e.

b) Find the x value(s) when or .

c) Determine the stationary points.
d) Apply either First derivative test or Second derivative test.
e) Conclusion from the test.

10.1 Extremum Problems Page 4

Example 3 (a)

Saturday, May 02, 2020 10:20 PM

By using first derivative test, find the relative extremum points for

Solution:

Stationary points :
First Derivative Test

test value

Tangent / /

Relative maximum point:
Relative minimum point:

Alternative:
• Usually apply for functions that are continuous for
such as polynomial functions.

Interval
test value

Tangent

Relative maximum point:
Relative minimum point:

10.1 Extremum Problems Page 5

Example 3 (b)

Saturday, May 02, 2020 10:20 PM

By using second derivative test, find the relative extremum points
for
Solution:

Stationary points :
Second Derivative Test
For

[maximum exists at
Relative maximum point:
For

[minimum exists at
Relative minimum point:

10.1 Extremum Problems Page 6

Point of Inflection

Saturday, May 02, 2020 2:40 PM

 If a function is defined at and the concavitiy of changes at

, then the point is called as inflection point .

Remarks:
An inflection point may NOT be a stationary point.

Steps to determine inflection points
a) Find the first derivative of the function, i.e.

b) Find the second derivative of the function, i.e.

c) Find the point(s) when or .

d) Use 2nd derivative to discuss concavity to determine inflection points.

10.1 Extremum Problems Page 7

Example 1

Saturday, May 02, 2020 10:20 PM

For the function , describe the concavity of the graph

and find if exist, the point of inflection.

Solution:

Interval
test value

Concavity

The function concaves downwards in the interval and

concaves upwards in the interval .

There exist point of inflection at

Inflection point :

10.1 Extremum Problems Page 8

Example 2

Saturday, May 02, 2020 10:20 PM

For the function , describe the concavity of the

graph and find if exist, the point of inflection.

Solution:

Interval
test value

Concavity

The function concaves upwards in the interval and ,

and concave downwards in the interval .

There exist point of inflection at and

0

Inflection points : ,

10.1 Extremum Problems Page 9

Example 3

Saturday, May 02, 2020 10:20 PM

Determine the intervals of increase and decrease for
and sketch the graph.

Solution:

Interval
test value

Tangent

The function is increasing in the intervals and and is
.
decreasing in the interval and

Relative maximum point:
Relative minimum point:

Diagram Reference: https://www.desmos.com/calculator/g0mfv2vrqy

10.1 Extremum Problems Page 10

Example 4

Saturday, May 02, 2020 10:20 PM

Find the coordinate of the stationary points and determine their

natures for and sketch the graph

Solution:

Stationary points :
Interval

test value

Tangent

Inflection point:
Minimum point:

Diagram Reference: https://www.desmos.com/calculator/uptu2vpo3i

10.1 Extremum Problems Page 11

Optimization Problems

Saturday, May 02, 2020 5:10 PM

 The theory on maximum and minimum values of a function can
be used to solve certain optimization problems.

Example 1:
Find 2 positive numbers whose sum is 30 and whose product is
maximum.

Solution:
Let be the positive numbers

Let

When [ is maximum]

[maximum p exists]

When

The two positive numbers are both 15.

 Strategy for solving optimization problems (Shapes // Solids)
(I) Draw a rough diagram based on information given question.
(II) State the corresponding equation of the quantity to be optimized.
(III) Rewrite the equation in (II) as a function in term of one variable.
(IV) Find the first derivative of the function in (III).
(V) Solve the first derivative when it is equal to zero.
(VI) Perform second derivative test to identify maximum or minimum.
(VII) Provide answer according to the requirement of the question.

10.1 Extremum Problems Page 12

Example 2

Saturday, May 02, 2020 10:20 PM

A cuboid has the following dimensions,
,

.
Calculate the maximum volume of the cuboid.

Solution:

Volume for cuboid in cm3

[rejected] For

Because the cuboid in this question
does not exist if

[maximum exists at

For maximum volume,

The maximum volume is cm3.

10.1 Extremum Problems Page 13

Example 3

Saturday, May 02, 2020 10:20 PM

A cone has a circular base with radius r and the height of the cone is h.

Given that the slant height is , find the value of r and h if the

volume of the cone is to be maximized.

Solution:

Volume for cone in cm3

Relationship between r & h

[rejected]
For

[maximum volume exists at

Since
When the volume of the cone is maximized,

which implies the cone has a circular base with a radius of cm
and a height of 4 cm .

10.1 Extremum Problems Page 14