How to find the global maximum of a function f(x)?

Question

Can someone explain to me the process on how to find the global maximum of a function f(x)? If you have any references that give step by step instructions and background information, you'll get the best vote.

Answer 1

First figure out the derivative of the function and solve for when this is equal to zero (critical points) the global maximum will either be one of these points or a boundary point for the set you are considering.

Answer 2

First you should convince yourself that the function f(x) even has a global maximum. For example, f(x) = x has no global maximum. If however, your function is defined on a compact set (for example, a closed interval [a,b] ), then your function indeed has a maximum, and you can proceed as follows:

For simplicity, assume that f(x) is defined on the interval [a,b]. Either the max occurs at the endpoints, or the maximum is in the interior (or both).

If f(x) has a maximum at a point p in the interior, then the derivative of f at p must be 0. So, as a first step, figure out all the places where the derivative is 0. Evaluate f(p) at each of these points.

Next evaluate f at the endpoints, a, b. Compare all the values: f(a), f(b), and the f(p)'s. Which ever value is the largest is your global maximum.

If f(x) is not defined on something like a closed interval, then you have to be more careful. You must first show that f(x) is bounded. Then you can find all the places where the derivative is zero and compare the values.

Of course, some of these places are local minimums, etc. To test whether a point actually is a LOCAL maximum, note that a sufficient condition for this is that the second derivative be negative at that point.

In my experience, however, when you are trying to find global maximums, it's usually faster and easier to ignore the second derivative test. Just find all the critical points, all the boundary points, and compare the values of f at these points.

Answer 3

take the derivative and set the new expression equal to 0, im sure this is the last thing the teacher taught you before she asked you this question

Answer 4

you can finde it only if function is diminishing. if its not it is endless. ok?