Let R(x)=−x2+47x+52R(x)=−x2+47x+52 denote the revenue in thousands of dollars generated from the production of x units

First, it pays to compute the profit function, the marginal revenue function and the marginal cost function.

The profit function is:

Profit(x)=R(x)−C(x)=−x2+47x+52−8x2−18x−24=−9x2+29x+28.Profit(x)=R(x)−C(x)=−x2+47x+52−8x2−18x−24=−9x2+29x+28.

The marginal revenue function is found simply by taking the first order derivative of revenue with respect to x, to get:

MR(x)=R′(x)=−2x+47.MR(x)=R′(x)=−2x+47.

The marginal cost function is found simply by taking the first order derivative of cost with respect to x, to get:

MC(x)=C′(x)=16x+18.MC(x)=C′(x)=16x+18.

a) The cost at an output of x=6 is simply:

C(6)=8(6)2+18(6)+24=288+108+24=420.C(6)=8(6)2+18(6)+24=288+108+24=420.

The marginal cost of an output of x=6 is simply:

MC(6)=C′(6)=16(6)+18=114.MC(6)=C′(6)=16(6)+18=114.

b) The additional cost of producing 0.5 units beyond the 6th unit is (0.5)(114) = 57. That's just the marginal cost of production at 6 times the extra units of quantity to be produced.

Therefore, from this and the answer to Part a) we can compute a close estimate of the total cost of producing 6.5 units by summing the cost of producing 6 units, 420, and the cost of producing the 6.5th unit, 57. Therefore, the estimate of C(6.5) generated this way is 420 + 57 = 477.

c) One definition of the break even point is where

R(x)=C(x).R(x)=C(x).

or revenues equal costs.

The two functions given for revenue and cost are a bit unusual because they're quadratic. That leads to a quadratic profit function.

Another definition of the break even point is where the profit function is zero. To compute the break even point, in this example, is to find the roots of the quadratic profit function. There are several online tools for computing roots like this. A really common one is Wolfram Alpha.

The roots are, immediately, x=-7/9 and x=4. That means profits are 0 at x=-7/9 and x=4. We'll assume that a negative output doesn't make sense in this context, so x is always greater than or equal to 0. In that case, the break even point is x=4.

d) At the break even point x=4, marginal revenue is MR(4)=39, and the marginal cost is MC(4)=82. So at the break even point, marginal revenue is significantly less than marginal cost.

e) Inspection of the marginal revenue and marginal cost functions reveals, that for low values of x equal to or above 0, marginal revenue is greater than marginal cost. To see this, compute the marginal revenue and marginal cost at 0, i.e. MR(0)=47 and MC(0)=18.

47 is greater than 18, so it pays to increase output beyond x=0 as profits will increase as x is increased.

In addition, marginal revenue at the break even point x=4, MR(4)=39, is less than marginal cost at the break even point x=4, MC(4)=82. So it would pay to make sure output is below x=4, as profits will increase as x declines.

If output were increased beyond the break even point, in this specific case, profits would fall (rapidly) and stay negative.