91Ó°ÊÓ

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ f(x)=\frac{2}{x+5} ;(3,7),[-6,4],(-\infty, 0),(-5,+\infty),[-5,+\infty),[-10,-5) $$

Find the area of the largest rectangle having a perimeter of \(200 \mathrm{ft}\).

Find the critical numbers of the given function. $$ f(x)=x^{3}+7 x^{2}-5 x $$

Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ f(x)=\frac{-2}{x+3} $$

Find the critical numbers of the given function. $$ f(x)=\frac{x}{x^{2}-9} $$

Given the circle having the equation \(x^{2}+y^{2}=9\), find (a) the shortest distance from the point \((4,5)\) to a point on the circle, and (b) the longest distance from the point \((4,5)\) to a point on the circle.

A manufacturer can make a profit of \(\$ 20\) on each item if not more than 800 items are produced each week. The profit decreases 2 cents per item over 800 . How many items should the manufacturer produce each week in order to have the greatest profit?

A school-sponsored trip will cost each student \(\$ 15\) if not more than 150 students make the trip; however, the cost per student will be reduced 5 cents for each student in excess of 150 . How many students should make the trip in order for the school to receive the largest gross income?

Suppose a weight is to be held \(10 \mathrm{ft}\) below a horizontal line \(A B\) by a wire in the shape of a \(Y\). If the points \(A\) and \(B\) are \(8 \mathrm{ft}\) apart, what is the shortest total length of wire that can be used?

(a) Draw a sketch of the graph of the given function on the indicated interval; (b) test the three conditions (i), (ii), and (iii) of the hypothesis of Rolle's theorem and determine which conditions are satisfied and which, if any, are not satisfied; and (c) if the three conditions in part (b) are satisfied, determine a point at which there is a horizontal tangent line. $$ f(x)=1-|x| ;[-1,1] $$