Question:
Locate all relative maxima, relative minima and saddle points (if any) for the function
$$f(x,y)=y\sqrt{x} - y^2-2x+7y$$
My attempt:
$$f_x(x,y)=\frac{y}{2\sqrt{x}}-2$$
$$f_y(x,y)=\sqrt{x}-2y+7$$
For the critical point $(x_0, y_0)$, $f_x(x_0,y_0)=0$ and $f_y(x_0,y_0)=0$. Following this line of reasoning we can find one critical point $(x_0,y_0)=(1,4)$. I will not expand upon that.
My interest is in $f_x(x,y)$ being undefined. Have a look at this definition from Anton:
13.8.5 Definition A point $(x_0, y_0)$ in the domain of a function $f(x, y)$ is called a critical point of the function if $f_x(x_0, y_0) = 0$ and $f_y(x_0, y_0) = 0$ or if one or both partial derivatives do not exist at $(x_0, y_0)$.
$f_x(x,y)$ does not exist for $x=0$. So, let our second critical point be $(0, k)$. Now, how do I find the value of $k$? It seems to me that $(0, k)$ represents an infinite number of points. So, does the above function have an infinite number of critical points?
