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) = -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#.

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