91Ó°ÊÓ

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

Of all rectangles that have a perimeter of \(42 \mathrm{ft},\) find the dimensions of the one with the largest area. What is its area?

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. $$ f(x)=12+9 x-3 x^{2}-x^{3} $$

From a thin piece of cardboard 20 in. by 20 in., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume?

Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the \(x\) -values at which they occur. $$f(x)=2 x^{3} ; \quad[-10,10]$$

Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, \(R(x),\) and cost \(, C(x),\) are in dollars. $$ R(x)=5 x, \quad C(x)=0.001 x^{2}+1.2 x+60 $$

Raggs, Ltd., a clothing firm, determines that in order to sell \(x\) suits, the price per suit must be $$ p=150-0.5 x $$ It also determines that the total cost of producing \(x\) suits is given by $$ C(x)=4000+0.25 x^{2} $$ a) Find the total revenue, \(R(x)\). b) Find the total profit, \(P(x)\). c) How many suits must the company produce and sell in order to maximize profit? d) What is the maximum profit? e) What price per suit must be charged in order to maximize profit?

Find the absolute maximum and minimum values of each function over the indicated interval, and indicate the \(x\) -values at which they occur. $$f(x)=(x+3)^{2 / 3}-5 ; \quad[-4,5]$$

Find \(\Delta y\) and \(f^{\prime}(x) \Delta x\). Round to four and two decimal places, respectively. For \(y=f(x)=3 x-1, x=4,\) and \(\Delta x=2\)

Use \(\Delta y \approx f^{\prime}(x) \Delta x\) to find a decimal approximation of each radical expression. Round to three decimal places. \(\sqrt{8}\)