I am hoping someone can please explain a few lines of working to me regarding critical points. I am really confused as to what I am actually doing in each step.
I have the following function:
$$ \\f(x,y)= 2x^2-y^2+6y$$
I am asked to find the absolute maximum and minimum value of this function on the disc $ \\x^2+y^2\le 16$. I am fine with finding the critical points on the actual function by setting the partial derivative with respect to x and y equal to 0 and solving. However I am having trouble understanding how we find critical points on the boundary.
So my first question is, when I solve the equation for the circle for $x^2$ and substitute into $f(x,y)$ to obtain $ \\g(y)= 32-3y^+6y$, what am I actually doing here in terms of the region/graph? What is $g(y)$ showing me?
Next, why do I then go on to differentiate $g(y)$ and solve for critical points? In my textbook, it says I am solving for the absolute extrema on the range from $-4\le y \le4$. I do not understand this.
Lastly, I don't get how doing this is sufficient for checking for critical points on the boundary. What about the range of x values that also goes from -4 to 4?
Perhaps a better understanding of my first question will really help the following but anyone that can help explain this to me, I would really appreciate it thank you!!
*I understand this question can be approached with re-parametrization of the circle however I am looking to understand it with this specific method.
