Find two positive numbers whose sum is 4 and such that the sum of te cube of the first an the square of the second is as small as possible.

Thanks any help would be appreciated!

hi

let x and y be the two positive numbers ,

\(\displaystyle x+y = 4\) -- 1

let \(\displaystyle f(x)=x^3+y^2\)

\(\displaystyle =x^3+(4-x)^2\)

\(\displaystyle =x^3+x^2-8x+16\)

\(\displaystyle f'(x)=3x^2+2x-8=(3x-4)(x+2)\)

When f'(x)=0 , x=4/3 or x=-2

x=4/3 is a local minimum and can be verified by taking the second derivative .