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
Increasing & Decreasing Function:

A function
f ( x) is said to be increasing/decreasing at
xa
if in
the immediate vicinity of the point [a, f (a)] 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
xa
indicate that the
function is increasing at s; negative first derivative indicates it is
decreasing.
f (a)  0 : increasing function at x  a
f (a)  0 : decreasing function at 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/decrease
Monotonically .
Concavity & Convexity :
A function f ( x) is concave at x  a if in some small region
close to the point [a, f (a)] the graph of the function lies
completely below the tangent line.
A function is convex at x  a if in an area very close to [a, f (a)]
the graph of the function lies completely above the tangent
line. A positive second derivative at x  a denotes that the
function is convex at x  a ; a negative second derivative at
x  a denotes the function is convex at a. The sign if first
derivative is irrelevant for concavity.
f (a)  0 : f ( x) is convex at x  a
f (a)  0 : f ( x) is concave at x  a
Concavity & Convexity :
Concavity & Convexity :
Relative Extrema:
f (a)  O
f (a)  O
f (a)  O : Relative Minimum at x  a
f ( a)  O : Relative Maximum at 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.

f (a)  O or is undefined

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,
f (a)  O, the function is concave at a, and hence at a relative maximum
f (a)  O, the function is convex at a, and hence at a relative minimum
f (a)  O, the test is inconclusive
Note that if the function is strictly concave/convex there will
be only one maximum/minimum called a global
maximum/global minimum.
Optimization of a Function: Example
Example: Optimize f ( x)  2 x3  30 x2  126 x  59
(a) Find the critical points by taking the first derivative, setting it equal to zero
and solving it for x
f ( x)  6 x2  60 x  126
6( x2 10 x  21)  0
x 2  7 x  3x  21  0
x( x  7)  3( x  7)  0
( x  3)( x  7)  0
x  3,
x  7 Citical Points
Optimization of a Function: Example
(b) Test for concavity by taking the second derivative, evaluating it at the critical
points, and checking the signs to destinguish between a relative maximum and minimum
f ( x)  6 x2  60 x  126
f ( x)  12 x  60
At x  3
f (3)  12(3)  60  36  60  24  0
At x  7 f (7)  12(7)  60  84  60  24  0
Concave, relative maximum
Convex, relative minimum
The function is maximized at x  3 and minimized at x  7
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: Optimize f ( x)  x3 18x2  96 x  80
Optimization: Example
Optimize f ( x)  x3 18x2  96x  80
Step I: Find the critical values
f ( x)  3x2  36 x  96
3( x2 12x  32)  0
x 2  8x  4 x  32  0
x( x  8)  4( x  8)  0
( x  8)( x  4)  0
x  8, x  4 Citical Points
Optimization: Example
Step II:Test for concavity to determine relative maximum
or minimum
f ( x)  6 x  36
At x  4
f (4)  6(4)  36  24  36  12  0
At x  8 f (8)  6(8)  36  48  36  12  0
Concave, relative maximum
Convex, relative minimum
Step III: Check the inflection points
f ( x)  6 x  36  0
6( x  6)  0
x6
Inflaction Point
Optimization: Example
Step IV:Evaluate the function at the critical values and
inflection points.
At x  4
f (4)  x3 18(4)2  96(4)  80  80
(4,80)
Relative Maximum
At x  6 f (6)  x3 18(6)2  96(6)  80  80
(6,64)
Inflaction Point
At x  8 f (8)  x3 18(8)2  96(8)  80  48
(8,48)
Relative Minimum
Questions ?
Thanks:
For Your Patience !