Question

Using Calculus, find two numbers whose sum is 23 and whose product is a maximum.

[please explain process]

Answer 1

You want to maximize xy, where x + y = 23.

Now if x + y = 23, then y = 23 - x. Hence
xy = x(23-x)

Now maximize that function, which is a function of just one variable,
\[ f(x) = x(23-x) \]

Answer 2

Yes, above answer is basically how you do it. Basically you are finding the rate of change. Lastly, to maximize the function, you would derive that f(x) and set it equal to zero, then plug it back into the original equation, and compare the values.