The sum of two positive number is 8. Find the two numbers such that the sum of the cube of one number and the square of the other is a minimum.

Question

anonymous · Answer

$$x+y=8$$
$$x^3+y^2$$ is to be minimized. 
first equation tells us 
$$y=8-x$$ so we have 
$$f(x)=x^3+(8-x)^2$$ so minimize. need some calc i think

anonymous · Answer

$$f(x)=x^3+x^2-16 x+64$$ 
$$f'(x)=3x^2+2x-16$$ set equal zero, find the critical points etc
you good from here?

anonymous · Answer

of course a cubic polynomial has no minimum, but here we have the domain (0,8) so there will be no doubt be a minimum in this interval