Why is $(0, 0)$ not a minimum of $f(x, y) = (y-3x^2)(y-x^2)$? There is an exercise in my lists about those functions:
$$f(x, y) = (y-3x^2)(y-x^2) = 3 x^4-4 x^2 y+y^2$$
$$g(t) = f(vt) = f(at, bt); a, b \in \mathbf{R}$$
It asks me to prove that $t = 0$ is a local minimum of $g$ for all $a, b
\in \mathbf{R}$
I did it easily:
$$g(t) = 3 a^4 t^4-4 a^2 t^2 b t+b^2 t^2$$
$$g'(t) = 2 b^2 t-12 a^2 b t^2+12 a^4 t^3$$
$$g''(t) = 2 b^2-24 a^2 b t+36 a^4 t^2$$
It is a critical point:
$$g'(0) = 0; \forall a, b$$
Its increasing for all a, b:
$$g''(0) = 2b^2 > 0; \forall b \ne 0$$
and
$$b = 0 \implies g(t) = 3 a^4 t^4$$
which has only one minimum, at $0$, and no maximum
However, it also asks me to prove that $(0, 0)$ is not a local minimum
of $f$.  How can this be possible? I mean, if $(0, 0)$ is a minimum over
every straight line that passes through it, then, in this point, $f$
should be increasing in all directions, no?

Draw the set of points in the $xy$-plane where $f(x,y) = 0$.  Then look at the regions of the plane that are created and figure out in which of them $f(x,y)$ is positive and in which $f(x,y)$ is negative.  From there, you should be able to prove that $(0,0)$ is neither a local maximum nor a local minimum point.  (Hint:  Using your sketch, is there any disk centered at $(0,0)$ in which $f(x,y)$ takes its minimum value at $(0,0)$?)
As important as understanding why the phenomenon you are observing is happening (no local minimum at the origin even though the function restricted to any straight line through the origin has a local minimum there) is to figure out how you would construct such a function and why the example given is, in some sense, the simplest kind of example you could construct.  The idea used here is very similar to creating an example which shows that $\lim_{(x,y) \rightarrow (0,0)} f(x,y)$ may not exist even if the limit exists along all straight lines approaching the origin.
What you're really learning in this example is that the seemingly reasonable intuition that we would naturally have that we can understand a function of two variables near a point by understanding all of the functions of one variable obtained by restricting the function to lines through that point is misguided -- in particular, it fails if we are trying to find local maxima and minima.
