1
$\begingroup$

I need to find find the maxima/minima of the function

$f(x,y) = 2x^4 - 3x^2y +y^2$, I have figured out that $(0,0)$ is the only critical point of this function.

But $rt-s^2 = \left( \frac{\partial^2 f}{\partial x^2}\right)\left( \frac{\partial^2 f}{\partial x^2}\right) - {\left(\frac{\partial^2 f}{\partial x\partial y}\right)}^2 = 0$ at $(0,0)$ and I am stuck at this point.

I tried to Change the function to $f(x,y) = 2\left({\left(x^2 -\dfrac{3y}{4}\right)}^2 - \dfrac{y^2}{16}\right)$ , to get some insight

But I am not sure How to proceed from here ?

Can anyone help me ??

Thank you

$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

Hint.

As

$$ 2 x^4 - 3 x^2 y + y^2 = (x^2 - y) (2 x^2 - y) $$

consider the change of variable

$$ y = \mu x^2 $$

$\endgroup$

You must log in to answer this question.