I need help with this problem. I have no idea where to start and how to get the answer.
Find three numbers such that the first is the sum of the second and third, the second is the square of the third, and the sum of the three numbers is a minimum.
Let the numbers be $\,x,y,z\,$:$$(1)\,\,x=y+z$$$$(2)\,\,y=z^2$$and we want the minimum of $$x+y+z=(y+z)+z^2+z=z^2+z+z^2+z=2(z+z^2)=:f(z)$$
Well, find the minimum of $\,f(z):\,\,f'(z)=2(1+2z)=0\Longleftrightarrow z=-1/2\,$ and etc.