Solving system of equations to find critical points of multivariable function

Question

I've been given the following function and asked to find all critical points.

$$f(x,y)=y^3+2x^2y+5xy$$

I began by finding partial derivatives $$\frac{\partial f}{\partial x}=4xy+5y$$ $$\frac{\partial f}{\partial y}=3y^2+2x^2+5x$$

I understand that to find critical points I need to equate both partial derivatives to $0$ and solve the system but I'm quite sure how to solve this particular system. Any tips?

Answer 1

Hint: $4xy+5y=0 \rightarrow y=0$ or $x=-\frac{5}{4}$. Now, consider these two cases and solve the second equation substituting $y$ or $x$. If $y=0$ you get $5x=0$ and so forth.

Answer 2

The first equation gives $$y(4x+5)=0.$$ Can you end it now?

I got: $$\left\{(0,0),(-2.5,0),\left(-\frac{5}{4},\frac{5}{2\sqrt6}\right), \left(-\frac{5}{4},-\frac{5}{2\sqrt6}\right)\right\}.$$