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A store sells \(Q\) units of a product per year. It costs a dollars to store one unit for a year. To reorder, there is a fixed cost of \(b\) dollars, plus \(c\) dollars for each unit. How many times per year should the store reorder, and in what lot size, in order to minimize inventory costs?

Find the absolute maximum and minimum values of each function, if they exist, over the indicated interval. Also indicate the \(x\) -value at which each extremum occurs. When no interval is specified, use the real line, \((-\infty, \infty)\). $$f(x)=x+\frac{3600}{x} ;(0, \infty)$$

Graph the function $$f(x)=\frac{x^{2}-3}{2 x-4}$$ a) Find all the \(x\) -intercepts. b) Find the \(y\) -intercept. c) Find all the asymptotes.

The total-cost and total-revenue functions for producing \(x\) items are $$ C(x)=5000+600 x \text { and } R(x)=-\frac{1}{2} x^{2}+1000 x $$ where \(0 \leq x \leq 600 .\) a) Find the total-profit function \(P(x)\) b) Find the number of items, \(x,\) for which the total profit is a maximum.

Total revenue, cost, and profit. Using the same set of axes. sketch the graphs of the total-revenue, total-cost, and total. profit functions. $$R(x)=50 x-0.5 x^{2}, \quad C(x)=4 x+10$$

Use your calculator's absolute-value feature to graph the follow. ing functions and determine relative extrema and intervals over which the function is increasing or decreasing. State the \(x\) -values at which the derivative does not exist. $$f(x)=\left|x^{2}-1\right|$$

Total revenue, cost, and profit. Using the same set of axes. sketch the graphs of the total-revenue, total-cost, and total. profit functions. $$R(x)=50 x-0.5 x^{2}, \quad C(x)=10 x+3$$

Assume the function \(f\) is differentiable over the interval \((-\infty, \infty)\) : that is, it is smooth and continuous for all real numbers \(x\) and has no corners or vertical tangents. Classify each of the following statements as cither true or false. If you choose false, explain why. If \(f\) has exactly two critical values at \(x=a\) and \(x=b\) where \(a

U.S. oil production. One model of oil production in the United States is given by $$ \begin{aligned} P(t)=& 0.0000000219 t^{4}-0.0000167 t^{3}+0.00155 t^{2} \\ &+0.002 t+0.22, \quad 0 \leq t \leq 110 \end{aligned} $$ where \(P(t)\) is the number of barrels of oil, in billions, produced in a year, \(t\) years after \(1910 .\) (Source: Beyond Oil, by Kenneth S. Deffeyes, p. \(41,\) Hill and Wang, New York, \(2005 .)\) a) According to this model, what is the absolute maximum amount of oil produced in the United States and in what year did that production occur? b) According to this model, at what rate was United States oil production declining in 2004 and in \(2010 ?\)

Assume the function \(f\) is differentiable over the interval \((-\infty, \infty)\) : that is, it is smooth and continuous for all real numbers \(x\) and has no corners or vertical tangents. Classify each of the following statements as cither true or false. If you choose false, explain why. The function \(f\) can have exactly one extreme value but no points of inflection.