Question

Find two numbers whose sum is 8 and whose product is a maximum?

Answer 1

Answer:

4

Step-by-step explanation:

Let the one number be x and another one as y

x + y = 32

the product of the numbers will be x × y

We can solve x + y = 8 for y.

y = 8 - x

x × y = x(8 - x) = 8x - x²

if we let the above function be f(x) , the f'(x) = 8 - 2x

the maximum value of this function will occur at a critical number.

A critical number occur where f'(x) = 0 or is undefined.

f'(x) is defined for all real x.

we only need to determine where f'(x) = 0

8 - 2x = 0

8 = 2x

x = 4

Answer 2

Given : Two numbers sum is 8 and   product is a maximum  

To Find : Two numbers

Solution:

Two numbers sum is 8

One number is x

Then other number is  8 - x

Product = x(8 - x)

P(x)  = 8x - x²

Taking derivative

P'(x) = 8 - 2x

 P'(x) = 0

8 - 2x = 0

=> x = 4

p''(x) = - 2  < 0

Hence product is maximum at x = 4

x = 4  => 8 - x = 4

Hence both the numbers are  4 and 4

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