Question

(6) Find maxima of f (x, y) = 2 (x² - y2) - x^4 + y^4 ​

Answer 1

hope it's helps.............

Answer 2

Find the first partial derivatives fx and fy.

fx(x,y) = 4x + 2y - 6

fy(x,y) = 2x + 4y

The critical points satisfy the equations fx(x,y) = 0 and fy(x,y) = 0 simultaneously. Hence.

4x + 2y - 6 = 0

2x + 4y = 0

The above system of equations has one solution at the point (2,-1).

We now need to find the second order partial derivatives fxx(x,y), fyy(x,y) and fxy(x,y).

fxx(x,y) = 4

fyy(x,y) = 4

fxy(x,y) = 2

We now need to find D defined above.

D = fxx(2,-1) fyy(2,-1) - fxy2(2,-1) = ( 4 )( 4 ) - 22 = 12

Since D is positive and fxx(2,-1) is also positive, according to the above theorem function f has a local minimum at (2,-1).

The 3-Dimensional graph of function f given above shows that f has a local minimum at the point (2,-1,f(2,-1)) = (2,-1,-6).