Problem statement: "What is the maximum product of two positive numbers whose sum is 96?"
Keyword: maximum, product, positive numbers, sum
sum: addition
product: multiplication
positive numbers: numbers greater than zero (which, by the way, is neither positive nor negative)
maximum: the greatest value; a function has, at most, one value of the dependent variable for each allowable value of the input variables.
If "I really don't get this" is related to the definitions, then review them." If it is a matter of finding a process to determine the answer, then read on.
The allowable input values (that is, the range) of this function is all positive numbers whose sum is 96. Right away, that limits the values. Since they must be "positive," that means each number must be greater than 0; since the sum must be 96 that means each value must be individually less than 96. This concept will be used often in later math, so understanding it now is vey important.
Now, of the various numbers that are greater than 0 and less than 96, the products (that is, the output or domain of this function) goes from just greater than 0 to some unknown value (that the problem wants us to find) and then back down to just greater than 0. The fact that the products do this is also an important concept, so, if you need to calculate them, find:
1 * 95 = ??
2 * 94 = ??
3 * 93 = ??
. . .
93 * 3 = ??
94 * 2 = ??
95 * 1 = ??
[Note: the products of the above expressions are duplicates, increasing and then decreasing. Since the problem stated "positive numbers," we must allow fractions even though I did not illustrate that.]
So, do you understand the problem, now? [note: re-read the above explanation if you need to because now I'm going to proceed to finding the answer.]
How to find the answer [algebra method]:
Look again at the list of products above (with answers increasing, then decreasing]. A important omitted answers are:
...
46 50 2300
47 49 2303
48 48 2304
49 47 2303
50 46 2300
...
Yep, the product increases to a maximum, then it decreases. Now, you must decide whether only integers are allowed and whether there is a maximum between two of the values listed. If the problem assumes that "numbers" means integers, then 48 and 48 are them. Now, we would certainly look between two pairs of values if their products were the same, but because this is only algebra, not calculus, we can "keep it simple."
So, put a note in your math diary/journal (you will need this for later math): "The maximum product of two positive numbers occurs at their mid-point."
Example of how to apply this rule: "If the sides of a square add up to x, what is the length of a side that gives the square the maximum area possible?"
[Note: if you take calculus, you will learn how to find formulas for the maximum and the minimum of complicated functions -- this is very handy, for example, when you must minimize the time a project takes, minimize the cost you must pay, maximize the profit you are going to earn, maximize the grade you are going to earn, ... A story: one day my high school son was standing in the kitchen talking about his summer job as a sales clerk. He said, "Dad, I know the prices just as well as any of the full-time people and I do well with customers, but full-time sales people earn twice as much as I do. If I earned twice as much per hour, then I'd only have to work half as many hours."]