There is a big difference between assuming that there is only one critical point and there is only one local maximum.
In the latter case counterexamples abound: a simple one is the function $f(x,y) = x^4 + y^4 - x^2 - y^2$. A critical point requires $4x^3 - 2x = 0$ and $4y^3 - 2y = 0$, and the only local maximum is at $(0,0)$, but the function is unbounded.
In the former case the only local maximum must be also a global maximum. This is a consequence of the Mountain Pass Lemma. (You can find an extremely high power version here; references to some finite dimensional version can be found on this MO discussion.)
The rough intuition is this:
- Label the local max as $x$. By definition as the local max if you move away from it a little bit the values go down.
- Suppose for contradiction there is another point $y$ with value above the local max that we know about.
- Take any curve connecting $x$ and $y$, and look at the value of the function along this curve. It starts high at $f(x)$, decreases a bit, and goes back up to $f(y)$. So it must have a minimum. So for each curve there is a minimum.
- Take the maximum of the minima among all curves, this can be shown to be a saddle (critical) point. This contradicts the assumption that the local max $x$ is the only critical point.