Critical points, extrema, and saddle points of $$$f{\left(x,y \right)} = x^{2} y^{2}$$$

Find and classify the critical points of $$$f{\left(x,y \right)} = x^{2} y^{2}$$$.

The first step is to find all the first-order partial derivatives:

$$$\frac{\partial}{\partial x} \left(x^{2} y^{2}\right) = 2 x y^{2}$$$ (for steps, see partial derivative calculator).

$$$\frac{\partial}{\partial y} \left(x^{2} y^{2}\right) = 2 x^{2} y$$$ (for steps, see partial derivative calculator).

Next, solve the system $$$\begin{cases} \frac{\partial f}{\partial x} = 0 \\ \frac{\partial f}{\partial y} = 0 \end{cases}$$$, or $$$\begin{cases} 2 x y^{2} = 0 \\ 2 x^{2} y = 0 \end{cases}$$$.

The system has the following real solutions: $$$\left(x, y\right) = \left(0, y\right)$$$, $$$\left(x, y\right) = \left(x, 0\right)$$$.

Now, let's try to classify them.

Find all the second-order partial derivatives:

$$$\frac{\partial^{2}}{\partial x^{2}} \left(x^{2} y^{2}\right) = 2 y^{2}$$$ (for steps, see partial derivative calculator).

$$$\frac{\partial^{2}}{\partial y\partial x} \left(x^{2} y^{2}\right) = 4 x y$$$ (for steps, see partial derivative calculator).

$$$\frac{\partial^{2}}{\partial y^{2}} \left(x^{2} y^{2}\right) = 2 x^{2}$$$ (for steps, see partial derivative calculator).

Define the expression $$$D = \frac{\partial ^{2}f}{\partial x^{2}} \frac{\partial ^{2}f}{\partial y^{2}} - \left(\frac{\partial ^{2}f}{\partial y\partial x}\right)^{2} = - 12 x^{2} y^{2}.$$$

$$$\left(0, y\right)$$$ is a set of an infinite number of points, they cannot be classified.

$$$\left(x, 0\right)$$$ is a set of an infinite number of points, they cannot be classified.

Relative Maxima

No relative maxima.

Relative Minima

No relative minima.

Saddle Points

No saddle points.

Critical points that cannot be classified

$$$\left(x, y\right) = \left(0, y\right)$$$A

$$$\left(x, y\right) = \left(x, 0\right)$$$A