Question 79956: 40. The cost in millions of dollars for a company to manufacture x thousand automobiles is given by the function C(x) = 4x2-40x+225. Find the number of automobiles that must be produced to minimize cost.
Answer by doctor_who(15)   (Show Source):
You can put this solution on YOUR website! A quadratic equation (like the one you have here) forms a "U" or "n" shaped curve. For the "U" type, there is a minimum value at the bottom of the "U" and likewise for the "n" shape, there is a maximum value at the top of the "n". The actual shape of the graph (ie a "U" or an "n" shape) depends on the rest of the numbers in the equation.
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At the minumum (OR MAXIMUM) value of a function, a tangent drawn across the curve at that point will be flat. Which is another way of saying the gradient of the curve at that point is zero.
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How to find gradients of curves ? Take the first derivative (ie "differentiate it")
so C = 4x^2 - 40x + 225
dC/dx = 8x - 40
At the minimum point, dC/dx will equal zero, so :
0 = 8x - 40
So 8x = 40
Therefore x = 5 [thousand automobiles] (ANSWER)
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Incidentally, how do we know that this gives a minimum cost and not a maximum one ? If you differentiate the equation again you get d2C/dx2 = 8 which is positive. So it is a minimum.
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