Finding maxima and minima of $f(x,y)=x^4+y^4-2x^2$

Question

Finding maxima and minima of $f(x,y)=x^4+y^4-2x^2$

I tried studying this exam problem but I need help in understanding it.

I found $f_{x}=4x^3-4x \\
f_y=4y^3$

From this, I get the stationary points $A(0,0)$, $B(1,0)$ and $C(-1,0)$

$f_{xx}=12x^2-4 \\
f_{yy}=12y^2 \\
f_{xy}=0$

Now after computing $D(x,y)$ I got $D(A)=D(B)=D(C)=0$ which is inconclusive.

In the graph, it shows that the point $B$ and $C$ are local minimum.

I would really appreciate some good explanation because I have an exam soon and I would like some help.

Answer 1

Your function can be written as $$f(x,y)=(x^2-1)^2+y^4-1$$ and we have $$(x^2-1)^2+y^4-1\geq -1$$. And we can conclude that there is no maximum.($\infty$)