Find two numbers whose sum is 24 and whose product is as large as possible

Solution:

Maxima and minima are known as the extrema of a function.

Maxima and minima are the maximum or the minimum value of a function within the given set of ranges

Let one number be x.

Then, the other number be (24 - x).

Let P (x) denote the product of the two numbers.

Thus, we have:

⇒ (x) = x (24 - x)

= 24x - x²

Therefore,

P' (x) = 24 - 2x

P" (x) = - 2

Now,

P' (x) = 0

⇒ 24 - 2x = 0

⇒ 24 = 2x

⇒ x = 12

Alsoo,

P" (12) = - 2 < 0

By the second derivative test, x = 12 is the point of local maxima of P.

Thus, the numbers are 12 and (24 - 12) = 12

Hence, the product of the numbers is the maximum when the numbers are 12 each

NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.5 Question 13

Find two numbers whose sum is 24 and whose product is as large as possible

Summary:

The two numbers whose sum is 24 and whose product is as large as possible are 12 and 12. Maxima and minima are known as the extrema of a function.