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 .