91Ó°ÊÓ

In Exercises 85 - 87, determine whether the statement is true or false. Justify your answer. The graph of a rational function can never cross one of its asymptotes.

In Exercises 87 - 94, use Descartes Rule of Signs to determine the possible numbers of positive and negative zeros of the function. \( g(x) = 2x^3 - 3x^2 - 3 \)

Briefly explain how to check polynomial division, and justify your reasoning. Give an example.

The profit \( P \) (in millions of dollars) for a recreational vehicle retailer is modeled by a quadratic function of the form P = at^2 + bt + c where \( t \) represents the year. If you were president of the company, which of the models below would you prefer? Explain your reasoning. (a)\( a \) is positive and \( -b / (2a) \le t \). (b)\( a \) is positive and \( t \le -b/(2a) \). (c)\( a \) is negative and \( -b /(2a) \le t \). (d)\( a \) is negative and \( t \le -b(2a) \).

In Exercises 93 - 96, use the Intermediate Value Theorem and the table feature of a graphing utility to find intervals one unit in length in which the polynomial function is guaranteed to have a zero. Adjust the table to approximate the zeros of the function. Use the zero or root feature of the graphing utility to verify your results. \( h(x) = x^4 - 10x^2 + 3 \)

(a) Graph \( y = ax^2 \) for \( a = -2, -1, -0.5, 0.5, 1 \) and \( 2 \). How does changing the value of affect the graph? (b) Graph \( y = (x - h)^2 \) for \( h= -4, -2, 2, \) and \( 4 \). How does changing the value of \( h \) affect the graph? (c) Graph \( y = x^2 + k \) for \( k = -4, -2, 2, \) and \( 4 \). How does changing the value of \( k \) affect the graph?

In Exercises 105 - 107, determine whether the statement is true or false. Justify your answer. A fifth-degree polynomial can have five turning points in its graph.

Sketch a graph of the function given by \( f(x) = x^4 \). Explain how the graph of each function \( g \) differs (if it does) from the graph of each function \( f \).Determine whether \( g \) is odd, even, or neither. (a) \( g(x) = f(x) + 2 \) (b) \( g(x) = f(x + 2) \) (c) \( g(x) = f(-x) \) (d) \( g(x) = -f(x) \) (e) \( g(x) = f\left(\frac{1}{2} x\right) \) (f) \( g(x) = \frac{1}{2}f(x) \) (g) \( g(x) = f(x^{\frac{3}{4}}) \) (h) \( g(x) = (f \circ f)(x) \)

A company that produces MP3 players estimates that the profit \( P \) (in dollars) for selling a particular model is given by \( P = - 76x^3 + 4830x^2 - 320,000, 0 \le x \le 60 \) where \( x \) is the advertising expense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of \( \$2,500,000 \)

A company that manufactures bicycles estimates that the profit \( P \) (in dollars) for selling a particular model is given by \( P = -45x^3 + 2500x^2 - 275,000, 0 \le x \le 50 \) where \( x \) is the advertising expense (in tens of thousands of dollars). Using this model, find the smaller of two advertising amounts that will yield a profit of \( \$800,000 \).