My attempt:
Let $$z = (x+y)^4 \iff \pm \sqrt[4]{z} = (x+y)$$
Since $z$ is a constant, $\sqrt[4]{z}$ will also be a constant.
$$(x+y) = \sqrt[4]{z} = c \iff y = -x + c$$
So the contour plot will look something like this:
The question also says, "From your sketch, determine whether $f$ has a maximum, a minimum or neither at the point (0,0)".
My answer would be "neither", since the point (0,0) is just a single point - it is not a part of a line or curve.