# Question about multivariable critical points

$$f(x, y) = x^4 + y^4 - 4xy + 3$$ I solved it and I got two minimum points $$(1,1)$$ and $$(-1,-1)$$ and I got $$(0,0)$$ as my saddle point but I'm suspecting that the function doesn't have a saddle point are my answers right?

• Wolfram Alpha agrees with your computation (as opposed to your suspicion). – Theo Bendit Oct 14 '19 at 5:23

Let be $$y=x$$ then
$$g(x) = f(x,x) = - 4x^2 +2x^4 + 3$$ which has a maximum at $$x=0$$. Let be $$y=-x$$, then $$h(x) = f(x, - x) = 4x^2 +2x^4 + 3$$ which has a minimum at $$x=0$$. Therefore your function does have a saddle point at $$(0,0)$$.