Question

classify all critical points
k(x, y) = x^(2) − xy + 4y^(2)

Answer 1

gradient of k(x,y) is
{2 x - y, -x + 8 y}
See when it is equal to {0,0} and you get a critical point

Answer 2

2 x - y=0
-x + 8 y=0 will get us
x=y=0
So we have to study the critical point (0,0)

Answer 3

that makes sense! thank you :)

Answer 4

You have now to study the Hessian Determinant at (0,0)

Answer 5

The hessian determinant is always 15 for this function so we need to study the second derivative

Answer 6

The second derivative of k \(x,y0 with respect to x is always 2, so the point (0,0) is a minimum

Answer 7

Here is a plot of your function