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

Answers (1)

Let x and y are two numbers
It is given that
x + y = 24 , y = 24 - x
and product of xy is maximum 
let f(x) = xy=x(24-x)=24x-x^2\\ f^{'}(x) = 24-2x\\ f^{'}(x)=0\\ 24-2x=0\\ x=12
Hence,  x = 12 is the  only critical value
Now,
f^{''}(x) = -2< 0
at x= 12  f^{''}(x) < 0
Hence, x = 12 is the point of maxima
Noe, y = 24 - x
            = 24 - 12 = 12
Hence, the value of x and y are 12 and 12 respectively
 

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