# 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)?

Dec 10, 2016

$50 , 50$

#### 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 $x \left(100 - x\right) = - {x}^{2} + 100 x$.

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

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

$- {x}^{2} + 100 x = - \left({x}^{2} - 100 x\right)$

$= - \left({x}^{2} - 100 x\right) - {\left(\frac{100}{2}\right)}^{2} + {\left(\frac{100}{2}\right)}^{2}$

$= - \left({x}^{2} - 100 x\right) - 2500 + 2500$

$= - \left({x}^{2} - 100 x + 2500\right) + 2500$

$= - {\left(x - 50\right)}^{2} + 2500$

Thus the vertex is at $\left(x , f \left(x\right)\right) = \left(50 , 2500\right)$, 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 $50 , 50$.