What is the difference between maximum and minimum in some applications? Just like for this: minimum 1.What are the two positive numbers whose product is 64 and whose sum is minimum ? While in maximum 2.What are the two positive numbers whose product is 64 and sum is maximum. How should these be answered ?

Minimum:

Let the two numbers be x and y.

ab = 64

Thus b = 64/a

And you are told that a + b is minimum

Which is a + 64/a is a minimum

Treat this expression as a curve. Just like most curves, this has a minimum.

So, let's say the curve is y = x + 64/x [Note that I substituted the a by x just for graphing purposes]

You can now either graph this, or find the derivative (if you've done that) to get the minimum point.

Though you don't have any maximum with this kind of problem, because there's infinite possibilities. Unless you remove the 'positive integer' requirement and you add an 'absolute' with the sum, and you will get two possibilities.