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)?

1 Answer
sente
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(100−x)=−x2+100x.

As f(x)=−x2+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:

−x2+100x=−(x2−100x)

=−(x2−100x)−(1002)2+(1002)2

=−(x2−100x)−2500+2500

=−(x2−100x+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.

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