Please, can someone give me more information on how to check if points are local or global maximum/minimum. I am aware of the second derivative test of determining the local minimum/maximum. But how do I check for global min/max? An example to illustrate this would be very much appreciated. Thanks.
[Math] Global maximum/ minimum of a function of more than one variable.
multivariable-calculus
Related Solutions
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:
plots[animate](plot,
[x^2+y^2*(1-x)^3, x = 0 .. 2, view = [0 .. 2, -1 .. 2]], y = 0 .. 10);
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:
There isn't a global maximum or minimum, at least not when considered on $\mathbb{R}^{2}$. Set $y=0$ and you can easily see that your function neither has an upper nor lower bound.
Related Question
- [Math] Local minimum and maximum of two variable implicit function
- How to classify critical points for a 2 variable function
- [Math] Finding the Maximum and Minimum Values of a Function in a Domain
- Local Maximum Point; Global Maximum Point
- Finding global maximum of a 2-variable function
- Finding Extrema points of a two variable function with parameters. (And a question about Global minimum).
Best Answer
A very useful method is Sturm's principle which says that a continuous function on a compact domain (e. g. a closed interval) which is bounded below/above an has at most one local minimum/maximum must have a global minimun/maximum at this very place.
If your function in question, however, has more than one local minimum/maximum, you have to compare the values of the function at all these minima/maxima and those with minimal/maximal value among these candidates are your global minima/maxima.
In each case, you should check first, that your function is bounded below/above!
Thus, for $f:[a,b] \rightarrow \Bbb R$ differentiable, you should proceed as follows: