Thinking about the shape of the graph (no calulus needed), what is the largest
value of f(x,y) = 1/(1+x^2+y^4)?
Well this one is easy, is obviously that f(0,0) = 1 is the highest value.
Otherwise if x or y increase in any direction from the origin (positive or negative), f will decrease exponenttialy.
Assuming x and y are real numbers, x^2 and y^4 are nonnegative, so the largest possible value is clearly f(x,y)=1 at (x,y)=(0,0) as draganicimw observed. One rather trivial correction - f does not decrease exponentially away from the origin; its decrease is only polynomial.