91Ó°ÊÓ

Let \(R(x), C(x),\) and \(P(x)\) be, respectively, the revenue, cost, and profit, in dollars, from the production and sale of \(x\) items. If $$ R(x)=5 x \text { and } C(x)=0.001 x^{2}+1.2 x+60, $$ find each of the following. a) \(P(x)\) b) \(R(100), C(100),\) and \(P(100)\) c) \(R^{\prime}(x), C^{\prime}(x),\) and \(P^{\prime}(x)\) d) \(R^{\prime}(100), C^{\prime}(100),\) and \(P^{\prime}(100)\) e) Describe what each quantity in parts (b) and (d) represents.

Find any relative extrema of each function. List each extremum along with the \(x\) -value at which it occurs. Then sketch a graph of the function. $$ f(x)=x^{2}+4 x+5 $$

Of all numbers whose difference is \(6,\) find the two that have the minimum product.

For the demand function given, find the following. a) The elasticity b) The elasticity at the given price, stating whether the demand is elastic or inelastic c) The value(s) of \(x\) for which total revenue is a maximum (assume that \(x\) is in dollars) $$ q=D(x)=500-2 x ; \quad x=57 $$

For the demand function given, find the following. a) The elasticity b) The elasticity at the given price, stating whether the demand is elastic or inelastic c) The value(s) of \(x\) for which total revenue is a maximum (assume that \(x\) is in dollars) $$ q=D(x)=\frac{400}{x} ; \quad x=50 $$

For each function, find all relative extrema and classify each as a maximum or minimum. Use the Second-Derivative Test where possible. $$ f(x)=-4 x^{2}+3 x-1 $$

Maximize \(Q=x y^{2}\), where \(x\) and \(y\) are positive numbers such that \(x+y^{2}=1\).

Maximize \(Q=x y,\) where \(x\) and \(y\) are positive numbers such that \(\frac{4}{3} x^{2}+y=16\).

Demand for chocolate chip cookies. Good Times Bakers determines that the demand function for its chocolate chip cookies is $$ q=D(x)=967-25 x $$ where \(q\) is the quantity of cookies sold when the price is \(x\) cents per cookie. a) Find the elasticity. b) At what price is elasticity of demand equal to \(1 ?\) c) At what prices is elasticity of demand elastic? d) At what prices is elasticity of demand inelastic? e) At what price is the revenue a maximum? f) At a price of 20 \& per cookie, will a small increase in price cause total revenue to increase or decrease?

Determine the horizontal asymptote of each function. If none exists, state that fact. $$ f(x)=\frac{2 x}{3 x^{3}-x^{2}} $$