1. Mathematical Economics
Concavity and Convexity
Relative Extrima
Inflection Points
Optimization of Functions
Optimization

2. Presented by: Nadia Batool Roll No. 8528 Rabia Naseer Roll No. 8503
Madeeha Iqbal Roll No. 8523
Mumtaz Hussain Roll No. 8506

3. 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.

4. 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 .

5. Concavity & Convexity :
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.

6. Concavity & Convexity :

7. Concavity & Convexity :

8. Relative Extrema:

9. 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.

10. Inflection Points:

11. 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

12. 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.

13. Optimization of a Function: Example

14. Optimization of a Function: Example

15. 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:

16. Optimization: Example

17. Optimization: Example

18. Optimization: Example

19. Questions ?

20. Thanks:
For Your Patience !