Which are the two numbers whose sum is 20 and the product is the maximum.
Let us assume one of the numbers is x. As the sum of the two numbers is 20, the other number is 20-x
The product of the two numbers is a function f(x) = x*(20-x) = 20x- x^2.
Now we need to find the maximum value of f (x). For this we need the derivative of f(x) and have to equate it to 0.
f’(x) = 20 – 2x = 0
=> 10 – x =0
=> x =10.
Now f’’(x) = -2 which is negative at x=10. So f (10) is truly the maximum value.
If x=10, the first number is 10 and the second number is 20-10 = 10.
So the two required numbers are 10 and 10.
Let the numbers whose sum is 20 be x and 20-x.
Then the product of the nimbers x and 20-x is P(x) = x(20-x).
P(x) = 20x-x^2
For the maximum , P'(c) = 0 and p"(c) <0.
P'(x) = ((20x-x^2)' = 20-2x
P'(c) = 0 for 20-2x = 0. Or for x =10.
P"(x) = (20-2x)' = -2. So P'(10 ) -2 which is less than zero.
Therefore for x = 10, the product P(x) = 10(20-10) = 100 is the maximum.