Question

Among all pairs of numbers whose sum is 100, how do you find a pair whose product is as large as possible. (Hint: express the product as a function of x)?

Answer 1

`50, 50`

Explanation:

Suppose two numbers sum to equal `100`. Let `x` represent the first number. Then the second number must be `100-x`, and their product must be `x(100-x) = -x^2+100x`.

As `f(x)=-x^2+100x` is a downward opening parabola, it has a maximum at its vertex. To find its vertex, we put it in vertex form, that is, `a(x-h)^2+k` where `(x,f(x))=(h, k)` is its vertex.

To put it into vertex form, we use a process called completing the square:

`-x^2+100x = -(x^2-100x)`

`=-(x^2-100x)-(100/2)^2+(100/2)^2`

`=-(x^2-100x)-2500+2500`

`=-(x^2-100x+2500)+2500`

`=-(x-50)^2+2500`

Thus the vertex is at `(x, f(x)) = (50, 2500)`, meaning it attains a maximum of `2500` when `x=50`.

As such, the pair of numbers `x, 100-x` attains a maximal product when `x=50`, meaning the desired pair is `50, 100-50`, or `50, 50`.