Solve the problem.
-The following polynomial approximates the rabbit population in a particular area. $R ( x ) = - 0.121 x ^ { 5 } + 3.133 x ^ { 4 } + 4500$, where $x$ is the number of years from year 0 . Use a graphing calculator to describe the rabbit population from the years 0 to 25 .
A) The population remains stable.
B) The population decreases.
C) The population increases.

Question

Quizplus · Accepted Answer

The polynomial $R(x) = -0.121x^5 + 3.133x^4 + 4500$ has a positive leading term ($3.133x^4$) and a negative term of higher degree ($-0.121x^5$). Initially, the positive term will dominate, causing the population to increase. However, as $x$ (years) increases, the negative term ($-0.121x^5$) will eventually dominate, but within the range of 0 to 25 years, the population is more likely to increase before any potential decrease due to the high initial value (4500) and the nature of polynomial growth.

Solve the Problem R(x)=−0.121x5+3.133x4+4500R ( x ) = - 0.121 x ^ { 5 } + 3.133 x ^ { 4 } + 4500R(x)=−0.121x5+3.133x4+4500

Solve the Problem $R ( x ) = - 0.121 x ^ { 5 } + 3.133 x ^ { 4 } + 4500$