I have actually two questions here, but both are very much related so I decided to put them both in this question.
From Wikipedia I found the following example of a function that has a single critical point which is a local minimum, yet is not a global minimum:
$f(x,y) = x^2 + y^2(1-x)^3$
Indeed the only critical point of $f$ is $(0,0)$ which is a local minimum by the second derivative test, but $(0,0)$ is not a global minimum because $f(4,1) = -11$. Why does this happen? I am having some trouble understanding intuitively why this is true. I know that $f$ has a minimum when restricted to a compact set. Then $(0,0)$ is a global minimum in any closed ball around $(0,0)$, and thus $(0,0)$ is a global maximum?? This is not true because $f$ is a counterexample, but I can't quite understand why.
In some cases we do have a global minimum at the unique critical point. For example, let
$f(x,y) = (-x+y)^2 + (x-1)^2 + (x+y-1)^2$
Then $f$ has a unique critical point at $(\frac{2}{3}, \frac{1}{6})$. The critical point turns out to be a global minimum (according to the graph and wolframalpha). How would I prove that the point is a global minimum? And how does one in general decide that a function has a global minimum/maximum on a critical point when the function is defined on all of $\mathbb{R}^2$, particularly when there is only one critical point?
Best Answer
Your intuitive picture is probably as follows: Imaging filling water into the hollow at the origin. As the water rises, since there are points lower than the origin, you expect the water to start spilling over into the lower lying areas at some point. And that point would need to be a critical point! The way to resolve the paradox is to note that before any spilling over occurs, the lake will in fact extend off to infinity. Look at a cross section of your function graph for fixed $y$, as $y$ starts at $0$ and then grows. In other words, study the function $x\mapsto x^2+y^2(1-x)^3$ as $y^2$ grows. You will see a local minimum, starting at the origin and approaching $x=1$ from below, and a local maximum, starting at infinity and approaching $x=1$ from above. As $y^2$ grows, the value at the local minimum grows, while the value at the local maximum decreases. So the aforementioned lake will extend out to $(1,\pm\infty)$ and then start to spill over the edge (the local maximum) at infinity before it spills over anywhere else.
A picture is tremendously helpful. If you have access to maple, try this:
Or you can take a look at this static plot of $f(x,y)$. It is easy to see how $(0,0)$ is a local, but not a global, maximum: