Find two positive numbers whose sum is 100 and whose product is a maximum.

Call the two numbers x and y. The constraint is that x + y =100, and we need to maximise A=xy.
Rearrange the constraint to y = 100 - x, and substitute into the product equation.
A = x(100-x) = 100x - x²
Differentiate to find the critical points:
A' = 100 - 2x = 0100 = 2xx = 50
Differentiate again to check that this is indeed a maximum.
A'' = -2
The second derivative is always negative so A = 100x - x² is always concave so the critical point is indeed a maximum.
Now it is easy to find y since we have x. y = 100 - 50 = 50
So the answer is x = 50 and y = 50.