mathematical economics
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Mathematical Economics. Concavity and Convexity Relative Extrima Inflection Points Optimization of Functions Optimization. Presented by:. Nadia Batool Roll No. 8528 Rabia Naseer Roll No. 8503 Madeeha Iqbal Roll No. 8523 Mumtaz Hussain Roll No. 8506. - PowerPoint PPT PresentationTRANSCRIPT
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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