Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum

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 (16 - x).

Let the sum of the cubes of these numbers be denoted by S (x).

Then,

S (x) = x³ + (16 - x)³

Therefore,

S' (x) = 3x² - 3(16 - x)²

S" (x) = 6x + 6(16 - x)

Now,

S' (x) = 0

⇒ 3x² - 3(16 - x)² = 0

⇒ x² - (16 - x)² = 0

⇒ x² - 256 - x² + 32x = 0

⇒ x = 256/32

⇒ x = 8

Also,

S" (8) = 6 (8) + 6 (16 - 8)

= 48 + 48

= 96 > 0

By the second derivative test, x = 8 is the point of local minima of S.

Thus, the numbers are 8 and (16 - 8) = 8

Hence, the sum of the cubes of the numbers is the minimum when the numbers are 8 each

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.5 Question 16

Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Summary:

The sum of the cubes of the numbers is the minimum when the numbers are 8 each. By the second derivative test, x = 8 is the point of local minima of S