Find two positive numbers x and y such that x + y = 60 and xy³ is maximum

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.

The two numbers are x and y such that x + y = 60

Therefore,

⇒ y = 60 - x

To find maxima or minima , first step is to differentiate and equate it to zero.

Let,

f (x) = xy³

f (x) = x (60 - x)³

Therefore,

On differentiating wrt x, we get

f' (x) = (60 - x)³ - 3x (60 - x)²

= (60 - x)² [60 - x - 3x]

= (60 - x)² (60 - 4x)

f' (x) = - 2 (60 - x)(60 - 4x) - 4 (60 - x)²

= - 2 (60 - x) [60 - 4x + 2 (60 - x)]

= - 2 (60 - x)(180 - 6x)

= - 12 (60 - x)(30 - x)

Now,

f' (x) = 0

⇒ x = 60 or x = 15

When, x = 60

Then,

f' (x) = 0

When, x = 15

Then,

f" (x) = - 12(60 - 15)(30 - 15)

= - 12 x 45 x 15 < 0

The second derivative is defined as the derivative of a function also known as double differentiation of a given function.

By the second derivative test,

x = 15 is a point of local maxima of f.

Thus, function xy³ is maximum when x = 15 and y = 60 - 15 = 45.

Hence, the required numbers are 15 and 45

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.5 Question 14

Find two positive numbers x and y such that x + y = 60 and xy³ is maximum.

Summary:

The two positive numbers x and y such that x + y = 60 and xy³ is maximum 15 and 45. Maxima and minima are known as the extrema of a function