## Maxima And Minima of Function of Two Variables

**Maxima And Minima of Function of Two Variables:**

**Maximum Value of Function:
**

A function f (x,y) is said to have a maximum point (a,b), if these exists a neighbourhood N of (a,b) such that;

**Minimum Value of Function:
**

A function f (x,y) is said to have a minimum point (a,b), if these exists a neighbourhood N of (a,b) such that;

**Necessary and Sufficient Conditions for Maxima and Minima:
**

The necessary conditions for a functions f (x,y) to have either a maximum and minimum at a point (a,b) are f_{x }(a,b) = 0 and f_{y} (a,b) = 0. The point (x,y) where x and y satiesfy f_{x }(x,y) = 0 and f_{y} (x,y) = 0 are called the Stationary or the Critical value of the functions.

Suppose (a, b) is a critical value of the functions f (x,y). Then f_{x }(a,b) = 0 and f_{y} (a,b) = 0. Now denote:

**1**. Then. the functions f (x,y) has maximum at (a,b) if AC - B^{2} > 0 and A < 0.

**2**. The functions f (x,y) has minimum at (a,b) if AC - B^{2} > 0 and A > 0.

Maximum and Minimum value of functions are called the **Extreme value of the function.**

**Working Rule to find the maximum and minimum value of function f (x,y):
**

**1**. Find fx (x,y) and fy (x,y).

**2**. Solve the equations fx (x,y) = 0 and fy (x,y) = 0.

**3**. Then find fxx (x,y), fxy (x,y) , fyy (x,y).

**4**. Then A = fxx (a,b) , B = fxy (a,b), C = fyy (a,b).

**5**. If AC - B2 > 0 and A < 0 the f has maximum at (a,b).

**6**. If AC - B2 > 0 and A > 0 the f has minimum at (a,b).

**7**. If AC - B2 < 0, then f has neither a maximum nor a minimum at (a,b) . The point (a,b) is called Saddle Point .

**8**. If AC - B2 = 0, further investigation is necessary.