Question

Find two positive real numbers whose prodcut is a maximum. 1) The sum is 10. 2) Sum of first and twice the second is 24. Please explain not just tell answer. Thanks

...?

Answer 1

This answer is for the first one.
The canonical form of a quadratic equation is
y=a(x-h)^2+k
where a is a constant,
(h,k) are the coordinates of the vertex.

Let x be one of the numbers, then 10-x is the other (sum=10). The product of the two numbers is therefore:
y=x(10-x)
=10x-x^2
Completing squares
=-1(x^2-10x+25) + 25
=-(x-5)^2+25
so the vertex is at (5,25),
which means that the numbers are x=5, and x=10-5=5 (both 5).