Answer to Question #258153 in Calculus for Latha

Question

Show that (0,0) is a critical point of f(x, y) = x ^ 2 + kxy + y ^ 2 no matter what value the

constant khas. (Hint: Consider two cases: k = 0 and k * 0.)

Answer 1

In the fisrt case, let k not equal to 0. So we get the following f_x(x,y)=2x+ky=0 and f_y(x,y)=kx+2y=0 therefore, f_x=0 implies y=-(2x)/(k) which also implies kx+2(-(2x)/(k))=kx-(4x)/(k)=x(k-(4)/((k)))=0

This implies x=0Vk=+-2 which also implies 

y=0Vy=+-x

In both cases you get the point (0,0) which is a critical point.

If we let k=0,

We get f(x,y)=x^(2)+y^(2) which implies f_x(x,y)=2x=0

f_y(x,y)=2y=0 implies that x=0 and y=0

In both cases, the critical point is (0,0)