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 March 3rd, 2012, 02:33 PM #1 Senior Member   Joined: Feb 2012 Posts: 110 Thanks: 0 Maximum production The average manufacturing cost per unit (in hundreds of dollars) for producing x units of a product is given by: C prime(x)= 2x^3-42x^2+288x+12 x is greater than or equal to 1 and less than or equal to 5 At what production level will the average cost per unit be maximum? Do I take the 1st and 2nd derivatives of this equation or do I just take the first?
March 3rd, 2012, 05:28 PM   #2
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Re: Maximum production

First, let's look at a plot of the function of the given interval:

[attachment=0:2v9icl33]avgcost.jpg[/attachment:2v9icl33]

Obviously the function has its maximum value at x = 5 since there appear to be no extrema on the given interval. Let's verify that:

$\frac{d}{dx}$$C'(x)$$=6x^2-84x+288=6(x-6)(x-" />

We see that indeed the function has no critical points on the given interval and also we see that the slope must be positive on that interval, so the maximum value must occur at the right endpoint of the interval, or x = 5.
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