Question

Find the critical points of f(x,y) = x^3 + y^3 + 3x^2 - 3y^2 - 8

Answer 1

You want it so that df/dx and df/dy both equal zero so:
\[3x ^{2} + 6x = 3y ^{2} - 6y = 0\]

which can be factorised to:

\[3x(x+2)=3y(y-2)=0\]

so x = 0, -2 and y = 0, 2

Answer 2

Right, so are there four critical points? (0,0), (0,2), (-2,0) and (-2,2)

Answer 3

Indeed they are

Answer 4

Thank you so much!