# Finding the minimum in a word problem by using derivatives

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.

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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.
You can either use derivatives, as proposed above (you asked for this), or else observe that $\,f(z)=2(z^2+z)\,$ is an upwards parabola and thus its vertex is a minimum point...anyway, above is written the point where the derivative vanishes. – DonAntonio Jun 11 '12 at 3:58
It's enough you find out the value for $\,z\,$ , as the other two values can be expressed in terms of it...*read carefully* my answer and try to understand it. – DonAntonio Jun 11 '12 at 4:03