Maximising and minimising $f(x,y)$ on $x^2+y^2\leqslant 9$

Question

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?

Answer 1

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.

Answer 2

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 $(0,-3)$.

Answer 3

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.

Answer 4

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.