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December 12th, 2014, 07:12 PM   #1
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Wink Can you help me find the number of automobiles that must be produced to minimize cost

O.K. so my math teacher says this is going to be on the final but we never did this so I don't know how to solve it:

The cost in millions of dollars for a company to manufacture x thousand automobiles is given by the function C(x)=4x^2-32x+128. Find the automobiles that must be produced to minimize the cost.

The answer is 4(thousand) but I don't know what to do to get that answer. Does anyone know?
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December 12th, 2014, 07:55 PM   #2
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To minimize a function, take the derivative and set it equal to zero. That, or one of the endpoints, should give the location of the minimum.
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December 12th, 2014, 08:42 PM   #3
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$$\begin{align*}C(x)&=4x^2-32x+128 \\ &= 4(x^2 - 8x + 32) \\ &= 4\left((x-8x+16)+16\right) \\ &= 4\left((x-4)^2+16\right) \end{align*}$$
And now we see that the minimum value of $(x-4)^2$ is zero - attained when $x = 4$. So the minimum value of $C(x)$ must be $4 \times 16 = 64$, attained when $x = 4$.

This is called 'completing the square' or putting the equation into 'vertex form'.

Last edited by v8archie; December 12th, 2014 at 08:47 PM.
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December 13th, 2014, 03:31 PM   #4
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Wow I must've actually learned something this semester I knew how to solve that. LOL.
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