# Among all pairs of numbers with a sum of 101, how do you find the pairs whose product is maximum?

##### 1 Answer

#### Explanation:

Suppose one of the numbers is

graph{x(101-x) [-100, 200, -5896, 4340]}

As this is a downward facing parabola, its maximum will occur at its vertex. Then, we can find the maximum by putting the quadratic into vertex form. We will do so using a process called completing the square.

Given the vertex form

From this, we know that

Intuitively this also makes sense, as the problem could also be interpreted as asking what side lengths create the rectangle with the greatest area given a fixed perimeter of