we have considered the problem of finding the relative extrema (relative maxima, relative minima) for a function of one independent variable. Now lets see the analogous problem of relative extrema for functions of two variables.

A function z = f(x,) is said to have a **relative maximum **at the point (a, b) if f(a, b) ≥ (x, y), for all (x, y) that are close to (a, b).

A relative maximum, or a relative minimum is referred to as a relative extremum.

We have used derivatives to find the relative maxima and relative minima of functions involving a single independent variable. For example, it was provide that if, y = f(x) has a relative maximum, or a relative minimum at x = a, then f ’(a) = 0. Similarly, if z = f(x,y) has a relative maximum, or a relative minimum at (a, b), then the two first order partial derivatives of z are zero at (a, b). That is

Fx (a,b) = 0, and fy (a,b) = 0

A point (a,b) for which f_{x} ( a,b) =f_{y} (a,b) = f_{y} (a,b) = 0 is called a **Critical point **of is called a **saddle point.**

Now, an outline on a test similar to the second derivative test as studied earlier is stated that gives conditions which determine whether a critical point yields a relative maximum, a relative minimum or a saddle point.

**Second Derivative Test For Relative Extrema**

For z = f(x,y)

Step-1 : Find fx (x,y) and fy (x,y)

Step – II. Find the critical points (S) by solving f_{x} (x,y) = 0 and fy (x,y) =0

Let (a, b) be a critical point.

Step III – Find f_{xx} (x, y) , fxy (x,y) and fyy (x,y) =0

Step Iv – Define A = f_{xx} (a,b), B = f_{xy} (a,b)

And C = f_{yy} (a, b)

Consider AC – B2. Then

(i) If AC – B^{2} > 0 and A < 0, f has a relative maximum at (a, b)

(ii) If AC – B^{2} >0 and A > 0, *f *has a relative minimum at (a, b).

(iii) If AC – B^{2}< 0. F has a saddle point at (a, b)

(iv) If AC – B^{2 }= 0, no conclusion can be drawn and further analysis is required.

It may be noted that when Ac –B^{2} > 0, the n neither A nor C can be zero. Further, both and C will be positive, or negative together.