Question

The sum of two natural numbers is 18. If the product of one number with the square of the

other is a maximum, find the numbers.

Answer 1

ANSWER : one natural number is $6$ the second is $12$ .

EXPLANATION

Let one number be $n\in N$ , then the second number is $(18-n)$ .To determine for which $n$ the product $n(18-n)^2$ will be maximum consider the function $f(x)=x(18-x)^2$ on the interval$[1,18]$ . Since $f'(x)=(18-x)^2-2x(18-x)=(18-x)(18-x-2x)=3(18-x)(6-x)$ and $(18-x)\geq 0$ on the interval $[1,18]$ , then $f'(x)>0$ if $x \in [1,6)$ and $f'(x)<0$ if $x\in (6,18)$ . Hence the function $f(x)$ increases on the interval $[1,6]$ and decreases on $[6,18]$

Thus, $f(1)<f(2)<...<f(6);$ $f(6)>f(7)>...>f(18)$ .

. So, $max _{[1,18]]}f(x) =f(6)$ . Therefore , one natural number is $6$ the second is $12$ .