## Extrema of Function of Several Variables

**Extrema of Function of Several Variables:**

**Definition:** We call f (a,b) a relative (local) maximum if there is an open Disk D centered at (a,b) for which f (a,b) ≥ f (x,y) for all (x,y) ∈ D. Similarly f (a,b) is called a relative (local) minimum there is an open Disk D for which f (a,b) ≤ f (x,y) for all (x,y) ∈ D. In either case f (a,b) is called a relative (local) extremum of f.

**Definition:** The point (a,b) is called a critical point of f if (a,b) is in the domain of f and either f = 0 or is undefined. Note that f = 0 if f_{x }(x,y) = 0 and f_{y} (x,y) = 0. f doesnot exist.

**Theorem:** If f (x,y) has a local extremum at (a,b) then (a,b) must be a critical point of f. However, if a point is a critical point it may not be a relative maximum or relative minimum. It could be saddle point. Look at (0,0) on the graph of z = x^{2} - y^{2}

But (0,0) is not a relative extremum of z.

**Theorem:** Suppose that f (x,y) has continuous 2nd order partial derivatives in some open disk containing the point (a,b) and that f_{x} (a,b) = f_{y} (a,b) = 0.

is called the Discriminant.

**Examples: Consider the functions f (x,y) = 16 - x ^{2} - y^{2}. Locate all critical point and classify them.**

**Solution: **

The only critical point is (0,0).

What about the functions f (x,y) = x^{2} y^{2}.

fx = 2x and fy = 2y and again the only critical point is (0,0).

Then f_{xx} = f_{yy} = 2 and fxy = 0.

At (0,0), D = 4 but f_{xx} > 0 ⇒ (0,0) is local minimum.