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