How do you prove an absolute maximum?

Answer 1

its the biggest of all relative maximums

Answer 2

You must check each possible maximum point and find the maximum which is the largest. If the function being tested tends to go to positive infinity, then the absolute maximum is + infinity.
A function may have several local maximum and minimum points, and then shoot off to positive infinity.

The sine wave is a good example; It has an absolute maximum of 1 at Pi/4 and an absolute minimum of -1 at 3Pi/4. This is the same over every 2Pi interval.

Simply taking the derivative and setting it to zero to find a max or min works fine for a quadrtic equation, but mor work is required if we are facing polynomials of higher degrees succh as 4th, 5th , or higher degrees ( meaning x^4, x^5, x^n).

Answer 3

It depends on what type of function you have. There may be a special case where its easy to show that f(x) <= f(5) for all x, or something.

However, the general method is:
If there is a maximum, either f'(x) = 0, or f'(x) is undefined (which includes all endpoints).
Thus you just need to test all those possible points, and take the biggest out of all of them.

Answer 4

Take the derivative of the function and set it equal to 0, solve for the variable...if the variable is positive, it is the maximum. If negative, it is the minimum.