Question

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

Answer 1

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