Question

r(x) = 0.05(x² + 1)(x - 6)
consider the graph of function r
use the graph to complete each statement.
function r has one complex zero and one real zero. based on the instantaneous growth rate, the population decr after 6 days and the population increased after 6 days.

Explanation:

Step1: Find real zeros of $r(x)$

Set $r(x)=0$: $0.05(x^2+1)(x-6)=0$.
$x^2+1=0$ has no real roots; $x-6=0$ gives $x=6$. So there is 1 real zero, and 2 complex zeros (from $x^2+1=0$, $x=\pm i$).

Step2: Analyze population change from graph

For $x<6$, the graph of $r(x)$ is below $y=0$, so population decreases. For $x>6$, the graph is above $y=0$, so population increases.

Answer:

Function $r$ has two complex zeros and one real zero. Based on the instantaneous growth rate, the population decreased before 6 days and the population increased after 6 days.