Consider fx y 3y12 2x2 Find and classify the critical poin

Consider f(x, y) = 3(y+1)2 - 2x2. Find and classify the critical points. Find the absolute maximum and absolute minimum of f in the domain bounded by the curves y = x and y = x2-6. Sketch the domain.

Solution

Given function f(x , y) = 3( y+ 1)^2 - 2x^2

To find the critical points of f(x , y)

First to find f_x = - 4x =0   ==> x = 0

       Find f _ y = 6( y+1)     = 0 ==> y = - 1.

There for critical point is ( 0 , -1).

f^2 _ x ( 0 , -1) = - 4 < 0 and f^2_y ( 0 , -1) = 6 > 0 .

So consider D( 0 , -1) = ( - 4)^2 + ( 6)^2 - 0   = 16 + 36 > 0,

There for   F(x , y) has relative maximun at the point ( 0 , -1) .   this is in the domain od the given curves y =x and   y = x^2 - 6.