4
Following up, we can tackle functions that are \"more different\" from quadratic
than problem 3 is. Try to find all critical points, and their nature for
$\frac{x^3}{3} + \frac{y^3}{3} - 3xy$
Hint There is more than one critical point. Try to locate and classify them all.
1 As discussed in an additional material file, if a function $f(x, y)$ has a
critical point at $(a, b)$, we can use the approximation $f(x, y) = f(a, b) +$
$\frac{1}{2}(f_{xx}(a, b)(x - a)^2 + 2f_{xy}(a, b)(x - a)(y - b) + f_{yy}(a, b)(y - b)^2) + E_{a, b}(x, y)$, where
$\lim_{(x, y)}\frac{E_{a, b}(x, y)}{(x - a)^2 + (y - b)^2} = 0$, so that, for $(x, y)$ close enough to $(a, b)$, the function $f$ should
essentially behave like the function you get ignoring the \"error term\" $E_{a, b}(x, y)$
