Location of maximum points of a solution to $y'=x+y^2$ In recent ODE test I faced with the following problem:
Suppose that integral curve (that is solution) of $y'=x+y^2$ has a maximum point $P= (p_x, p_y)$. Can it be located in


*

*Quadrant I?

*Quadrant II?

*Quadrant III?


I answered the following way:


*

*no

*yes

*yes


My reasoning: Since $y'(p_x) = 0$ then $p_x = -y^2$, so $p_x \le 0$.
However, my answer was wrong. Can you please point out my mistake and provide correct answer and solution?
Thanks a lot for your help!

If $y$ is a differentiable function and $x_0$ is a stationary point, $y'(x_0)=0$. Since we know that $y'(x_0)=x_0+y(x_0)^2$ and $y(x_0)^2\geq 0$, it follows that $x_0\leq 0$. Since $y''=1+2yy'=1+2y(x+y^2)$ we get that every time $x+y^2=y'$ equals zero the second derivative $y''$ equals one. In particular every stationary point is a local minimum, not a maximum.
