Maximising and minimising $f(x,y)$ on $x^2+y^2\leqslant 9$ 
Find the absolute minimum and maximum of $f(x,y):=x^2+y^2-8y+3$ on the
  disc $x^2+y^2\leqslant 9$.

I know what the answer is and how to obtain it. 
What I do not understand is why we may assume that the extrema occur on the boundary of the disc.
I first looked at $Jf(x,y)=\mathbf{0}$ which occurs iff $(x,y)=(0,4)$, which is outside the disc.
One can then use the Lagrange Multiplier Theorem with the constraint $x^2+y^2=9$ to obtain the correct answers.
But I do not understand how we know this must be the constraint. How can we know for certain that the minimum and maximum do not occur in the interior of the disc?
 A: You can  use a very elementary argument to argue that maximum and minimum must occur on the boundary. Suppose, if possible, the maximum occurs at a point $(a,b)$ with $a^{2}+b^{2} <9$. By increasing/decreasing $a$ slightly you can  stay within the disk but make the first term in $f$ larger. This contradiction shows that maximum occurs on the boundary. Similarly for the minimum. 
A: Thinking geometrically, note that $f(x,y) = x^2 + (y-4)^2 - 13$. So to minimize or maximize $f(x,y)$ in a region you want to minimize or maximize the distance from the point $P=(0,4)$.
The points in the disc $x^2+y^2 \le 9$ at minimum and maximum distances from $P$ will obviously be on the boundary of the disc - in fact, these points are $(0,3)$ and $(0,-3)$.
A: A local maximum/minimum for $f$ in an open region is a critical point, i.e. its gradient $\nabla f$ vanishes at that point.
However, the gradient of $\nabla f=2(x,y-4)$ so it only vanishes at $(0,4)$, which lies outside the disk $x^2+y^2\le9$. Therefore, if a minimum/maximum exists, it must occur on the boundary of the disk.
A: Since $y\geq-3$, we obtain:
$$x^2+y^2-8y+3\leq9-8\cdot(-3)+3=36.$$
The equality occurs for $x=0$ and $y=-3,$ which says that we got a maximal value.
Also, since $y\leq3$, we obtain:
$$x^2+y^2-8y+3=y^2-8y+3=-12+x^2+(3-y)(5-y)\geq-12.$$
The equality occurs for $x=0$ and $y=3$, which says that we got a minimal value.
