Question

the function c gives the production cost for a bakery to produce cakes of a certain type, where c(x) is the cost, in dollars, to produce x of the cakes. the function m defined by m(x)=c(x + 1)-c(x) gives the marginal cost, in dollars, to produce cake number x + 1. which of the following gives the best estimate for the marginal cost, in dollars, to produce the 40th cake? a \\( \frac{c(39)}{39} \\) b \\( \frac{c(40)}{40}-\frac{c(39)}{39} \\) c c(39) d c(40)-c(39)

Explanation:

Step1: Recall the definition of marginal cost

The marginal cost is the derivative of the cost function. The marginal cost to produce the \((x + 1)\) - th unit is approximated by the derivative of the cost function \(C(x)\) at \(x\).

Step2: Substitute \(x = 39\) for the 40 - th cake

Since we want the marginal cost to produce the 40 - th cake (i.e., when \(x+1 = 40\), so \(x = 39\)), we use \(C^{\prime}(x)\) at \(x = 39\). So the marginal cost to produce the 40 - th cake is \(C^{\prime}(39)\)

Options A and B are related to average cost (\(\frac{C(x)}{x}\) is the average cost function). Option D is the difference of derivatives which does not represent the marginal cost of the 40 - th cake.

Answer:

C. \(C^{\prime}(39)\)