Critical Points, Extrema, and Saddle Points Calculator
Find the critical points, extrema, and saddle points of a function
The calculator will try to find the critical (stationary) points, the relative (local) maxima and minima, as well as the saddle points of the multivariable function, with steps shown.
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Your Input
Find and classify the critical points of $$$f{\left(x,y \right)} = - 2 x^{2} + 2 x^{2 y} + y^{3} - 2 y^{2} + 2$$$.
Solution
The first step is to find all the first-order partial derivatives:
$$$\frac{\partial}{\partial x} \left(- 2 x^{2} + 2 x^{2 y} + y^{3} - 2 y^{2} + 2\right) = - 4 x + 4 x^{2 y - 1} y$$$ (for steps, see partial derivative calculator).
$$$\frac{\partial}{\partial y} \left(- 2 x^{2} + 2 x^{2 y} + y^{3} - 2 y^{2} + 2\right) = 4 x^{2 y} \ln\left(x\right) + 3 y^{2} - 4 y$$$ (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 $$$\begin{cases} - 4 x + 4 x^{2 y - 1} y = 0 \\ 4 x^{2 y} \ln\left(x\right) + 3 y^{2} - 4 y = 0 \end{cases}$$$.
As can be seen, there is no solution, so no critical points.
Answer
Relative Maxima
No relative maxima.
Relative Minima
No relative minima.
Saddle Points
No saddle points.