I've been very confused about this 1984 VTRMC problem and I tried drawing possible graph below, but I suspect that its wrong somehow. (With the way its positioned now, couldn't the local max get infinitely closer to $(x_0, f(x_0))$?
Any advice, please?
With the conditions $f(x_0)\ge f(x_0+h), f(x_0)\ge f(x_0-h)$, it is clear there must be a local maximum of $f$ inside $[x_0-h, x_0+h]$. There could be one or there could be many such local maximums.
The problem tasks you with finding an $r$ so you can be sure that at least one such local maximum point is inside or on the boundary of that circle. And it asks you to find that $r$ not knowing $f$ exactly, you only get the stated information about $f$, namely $h, M$ and $d$.
So while an example as yours, where there are local maximums converging to $(x_0, f(x_0))$ exist, those are not the only possible $f$.
Look at the following example for $M=2.0001, h=1, d=1:$
The only local maximum is at $(\frac23,\frac43)$ and the green circle does not contain it. Since $M$ is an (unknown, but given) constant, the maximum can't go to much to the right or left of $x_0$, as that would mean the function has to grow faster than $M$ or fall faster then $-M$, so increasing the radius of the green circle a little will help to catch all cases.
The problems asks you for the smallest $r$ only because in this case the red circle will obviusly also contain any local maximum point. By asking for the smallest $r$, it makes sure you give the best possible answer that works for all $f$ satisfying the conditons. For some of these functions the $r$ could be lowered more, but then we need to know more about that function, but we don't know more.