Find two positive numbers whose sum is 12 and their product is maximum

1.	Find two positive numbers whose sum is 12 and their product is maximum.
Answer» Let the two numbers be x, 12 – x. Their product p = x (12 – x) = 12x – x² To find the maximum product. p'(x) = 12 – 2x p”(x) = -2 p'(x) = 0 ⇒ 12 – 2x = 0 ⇒ 2x = 12 ⇒ x = 6 at x = 6, p”(x) = -2 = -ve ⇒ p is maximum at x = 6 when x = 6, 12 – x = 12 – 6 = 6 So the two numbers are 6, 6

Find two positive numbers whose sum is 12 and their product is maximum.

Answer»

Let the two numbers be x, 12 – x.

Their product p = x (12 – x) = 12x – x²

To find the maximum product.

p'(x) = 12 – 2x

p”(x) = -2

p'(x) = 0 ⇒ 12 – 2x = 0 ⇒ 2x = 12

⇒ x = 6

at x = 6, p”(x) = -2 = -ve

⇒ p is maximum at x = 6

when x = 6, 12 – x = 12 – 6 = 6

So the two numbers are 6, 6