A certain cost function has the following graph: a. The...

A certain cost function has the following graph: a. The associated marginal cost is (A) increasing, then decreasing (B) decreasing, then increasing (C) always increasing (D) always decreasing b. The marginal cost is least at approximately (A) $x=0$ $\begin{array}{lll}\text { (B) } x=50 & \text { (C) } x=100 & \text { (D) } x=150\end{array}$ c. The cost of 50 items is (A) approximately $\$ 20$, and increasing at a rate of about $\$ 3,000$ per item (B) approximately $\$ 0.50$, and increasing at a rate of about $\$ 3,000$ per item (C) approximately $\$ 3,000$, and increasing at a rate of about $\$ 20$ per item (D) approximately $\$ 3,000$, and increasing at a rate of about $\$ 0.50$ per item

A certain cost function has the following graph:
a. The associated marginal cost is
(A) increasing, then decreasing
(B) decreasing, then increasing
(C) always increasing
(D) always decreasing
b. The marginal cost is least at approximately
(A) $x=0$
$\begin{array}{lll}\text { (B) } x=50 & \text { (C) } x=100 & \text { (D) } x=150\end{array}$
c. The cost of 50 items is
(A) approximately $\$ 20$, and increasing at a rate of about $\$ 3,000$ per item
(B) approximately $\$ 0.50$, and increasing at a rate of about $\$ 3,000$ per item
(C) approximately $\$ 3,000$, and increasing at a rate of about $\$ 20$ per item
(D) approximately $\$ 3,000$, and increasing at a rate of about $\$ 0.50$ per item

Key Concepts

Cost Function

A cost function represents the total cost incurred to produce a given number of items. It combines both fixed and variable costs and is used in economics and business to analyze production efficiency and to make decisions about output levels.

Marginal Cost

Marginal cost is the derivative of the cost function with respect to quantity. It indicates the additional cost incurred when one more unit is produced and is essential for understanding how costs change with variations in production.

Derivative and Rate of Change

The derivative is a fundamental concept from calculus that measures the rate at which a function changes at any point. In the context of cost functions, it tells us how rapidly the cost increases or decreases as production levels change, which is critical for determining optimal production levels.

Graphical Analysis of Functions

Graphical analysis involves examining the shape and behavior of a function’s graph to extract information about its increasing or decreasing nature, inflection points, and local extrema. This analysis helps in understanding the relationship between a cost function and its marginal cost.

Local Extrema and Critical Points

Local extrema refer to points where a function reaches a minimum or maximum value locally. Identifying these points through analysis of the derivative (i.e., where it equals zero or changes sign) is key in optimization problems, such as finding the production level with the least marginal cost.

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