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 fx ( a,b) =fy (a,b) = fy (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 fx (x,y) = 0 and fy (x,y) =0
Let (a, b) be a critical point.
Step III – Find fxx (x, y) , fxy (x,y) and fyy (x,y) =0
Step Iv – Define A = fxx (a,b), B = fxy (a,b)
And C = fyy (a, b)
Consider AC – B2. Then
(i) If AC – B2 > 0 and A < 0, f has a relative maximum at (a, b)
(ii) If AC – B2 >0 and A > 0, f has a relative minimum at (a, b).
(iii) If AC – B2< 0. F has a saddle point at (a, b)
(iv) If AC – B2 = 0, no conclusion can be drawn and further analysis is required.
It may be noted that when Ac –B2 > 0, the n neither A nor C can be zero. Further, both and C will be positive, or negative together.