Mathematical EconomicsConcavity and ConvexityRelative ExtrimaInflection Points Optimization of FunctionsOptimization

Presented by:

Nadia Batool Roll No. 8528Rabia Naseer Roll No. 8503Madeeha Iqbal Roll No. 8523Mumtaz Hussain Roll No. 8506

Increasing & Decreasing Function:

A function is said to be increasing/decreasing at if in

the immediate vicinity of the point the graph of the function

rises/falls as it moves from left to right.

Since the first derivative measures the rate of change and slope of a

function, a positive first derivative at indicate that the

function is increasing at s; negative first derivative indicates it is

decreasing.

( )f x x a[ , ( )]a f a

x a

( ) 0 : increasing function at ( ) 0 : decreasing function at f a x af a x a

Increasing & Decreasing Function:

Monotonic Function: A function that increases/decreases over its entire domain is called monotonic function. It is said to increase/decreaseMonotonically .

A function is concave at if in some small region close to the point the graph of the function lies completely below the tangent line.

A function is convex at if in an area very close to

the graph of the function lies completely above the tangent

line. A positive second derivative at denotes that the function is convex at ; a negative second derivative at

denotes the function is convex at a. The sign if first derivative is irrelevant for concavity.

Concavity & Convexity :

( )f x x a[ , ( )]a f a

x a [ , ( )]a f a

x ax a

x a

( ) 0 : ( ) is convex at ( ) 0 : ( ) is concave at

f a f x x af a f x x a

Concavity & Convexity :

Concavity & Convexity :

Relative Extrema:

( ) ( ) : Relative Minimum at ( ) ( ) : Relative Maximum at

f a O f a O x af a O f a O x a

Inflection Points:

An inflection point is a point on the graph where the function

crosses its tangent line and changes from concave to convex

or vice versa. Inflection points occur only where the second

derivative equals to zero or is undefined. The sign of first

derivative is immaterial.

( ) or is undefinedf a O

Concavity changes at x a

Graph crosses its tangent line at x a

Inflection Points:

Optimization of a Function:

Optimization is the process of finding the relative maximum or minimum of a function. It is developed through usual differential functions

Step I: Take the first derivative, set it equal to zero, and solve for the critical points. This step represents the necessary condition know as the first-order condition. It identifies all the points at which the function is neither increasing nor decreasing, but at a plateau. All such points are candidates for a possible relative maximum or minimum

Optimization of a Function:

Step II: Take the second derivative, evaluate it at the critical points and check the signs. If at a critical point a,

Note that if the function is strictly concave/convex there will be only one maximum/minimum called a global maximum/global minimum.

( ) , the function is concave at a, and hence at a relative maximum ( ) , the function is convex at a, and hence at a relative minimum ( ) , the test is inconclusive

f a Of a Of a O

Example: 3 2Optimize ( ) 2 30 126 59f x x x x ( ) Find the critical points by taking the first derivative, setting it equal to zero and solving it for xa

2 ( ) 6 60 126f x x x 2 6( 10 21) 0x x

2 7 3 21 0x x x ( 7) 3( 7) 0x x x

( 3)( 7) 0x x

3, 7 Citical Points x x

Optimization of a Function: Example

Optimization of a Function: Example

( ) Test for concavity by taking the second derivative, evaluating it at the criticalpoints, and checking the signs to destinguish between a relative maximum and minimumb

( ) 12 60f x x

At 3 (3) 12(3) 60 36 60 24 0 Concave, relative maximumx f

2 ( ) 6 60 126f x x x

At 7 (7) 12(7) 60 84 60 24 0 Convex, relative minimumx f

The function is maximized at 3 and minimized at 7x x

Optimization:

Step I: Find the critical values

Step II: Test for concavity to determine relative maximum or

minimum

Step III: Check the inflection points

Step IV: Evaluate the function at the critical values and

inflection points.

Example: 3 2Optimize ( ) 18 96 80f x x x x

Optimization: Example3 2Optimize ( ) 18 96 80f x x x x

Step I: Find the critical values2 ( ) 3 36 96f x x x

2 3( 12 32) 0x x 2 8 4 32 0x x x

( 8) 4( 8) 0x x x ( 8)( 4) 0x x

8, 4 Citical Points x x

Optimization: Example

Step II:Test for concavity to determine relative maximum or minimum

( ) 6 36f x x At 4 (4) 6(4) 36 24 36 12 0 Concave, relative maximumx f

At 8 (8) 6(8) 36 48 36 12 0 Convex, relative minimumx f

Step III: Check the inflection points( ) 6 36 0f x x

6( 6) 0x 6 Inflaction Point x

Optimization: Example

Step IV:Evaluate the function at the critical values and inflection points.

3 2At 4 (4) 18(4) 96(4) 80 80 (4,80) Relative Maximumx f x 3 2At 6 (6) 18(6) 96(6) 80 80 (6,64) Inflaction Pointx f x 3 2At 8 (8) 18(8) 96(8) 80 48 (8,48) Relative Minimumx f x

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