Question: Among all pairs of numbers with a sum of 119, find the pair whose product is maximum.
Let the two numbers be x and y. Then,
Sum is S = x + y
Product is P = xy
So, P = x(S-x)
P = 119x - x^2
Let it be noted here that P is a quadratic function of x, i.e
P(x) = 119x - x^2 = 59.5^2 + 2*59.5*x - x^2 - 59.5^2 = 59.5^2 - (59.5 - x)^2
And obviously P(x) has a maximum value for x = 59.5
Since x = 59.5 and x + y = 119, then y = 59.5.
Answer: pair of numbers with maxm product is x = 59.5 and y = 59.5