Let fx1 x2 x21 2x1 x22 4x2 1 Find the maximum and minim

Let f(x_1, x_2) = x^2_1 - 2x_1 - x^2_2 + 4x_2 + 1. Find the maximum and minimum values of f when 0 Inequality x_1 Inequality 2 and 0 Inequality x_2 Inequality 2.

Solution

f(x1, x2) = x1^2 -2x1 -x2^2 +4x2 +1

find fx1 = 2x1 -2

find fx2 = -2x2 +4

set fx1 =0 and fx2=0 and solve to get x1 and x2

x1= 1 ; x2 = 2

(x1, x2) = (1,2)

Now find fx1x1 = 2

fx2fx2 = -2

fx1x2 = 0

So, D = fx1x1(1,2) *fx2x2(1,2) -fx1x2^2( 1, 2) = 2*(-2) - 0 = -4

If D < 0, then f has a saddle point at (1,2).no maxima or minima