Question

Considering the daily profit function p(x) = -0.0002x⁹ - 300x - 0.0003x² for the racket manufacturer, what is the significance of the term with the highest degree x⁹ in the context of the manufacturer's profit, and how does it influence the overall profit equation?

A)The term -0.0002x⁹ represents the long-term growth factor, indicating the most substantial impact on profit over time.

B) The term -0.0002x⁹ signifies the initial investment cost, contributing significantly to the overall profit.

C) The term -0.0002x⁹ is a negligible factor in the profit equation, with minimal impact on the manufacturer's overall profitability.

D) The term -0.0002x⁹ influences short-term profits, contributing substantially to the overall profit equation.

Answer 1

The term -0.0002x^9 dictates the end behavior of the profit function, indicating that at very high production levels, there are diminishing returns and potential losses.

Step-by-step explanation:

The term -0.0002x9 in the racket manufacturer's daily profit function p(x) = -0.0002x9 - 300x - 0.0003x2 is significant because it represents the leading term, which is the term with the highest degree in the polynomial. The leading term often dictates the end behavior of the function, that is, how the profit behaves for very large quantities of x (the number of units produced and sold). Because this term has a negative coefficient, as the quantity x gets very large, the profit p(x) will decrease sharply due to the negative ninth power. Thus, the term with the highest degree, in this case, suggests that the profit function experiences diminishing returns at higher output levels and can eventually lead to losses if the production quantity is too high.