Question

a populations instantaneous growth rate is the rate at which it grows for every instant in time. function r gives the instantaneous growth rate of a bacterial population x hours after the start of an experiment. r(x) = 0.05(x - 1)(x - 3)(x - 5) use the graphing tool to graph the function, and then complete the statements. the instantaneous growth rate is zero at hours after the start of the experiment. the population increases between hours after the experiment began. the population decreases between hours after the experiment began.

Explanation:

Step1: Find roots of $r(x)=0$

Set $0.05(x-1)(x-3)(x-5)=0$. By zero-product property, solve each factor:
$x-1=0 \implies x=1$; $x-3=0 \implies x=3$; $x-5=0 \implies x=5$

Step2: Analyze sign of $r(x)$

Test intervals:

• For $x<1$: $r(x)=0.05(-)(-)(-)=negative$
• For $1
• For $3
• For $x>5$: $r(x)=0.05(+)(+)(+)=positive$

Growth rate positive = population increases; negative = decreases.

Answer:

The instantaneous growth rate is zero at $\boldsymbol{1, 3, 5}$ hours after the start of the experiment.
The population increases between $\boldsymbol{15}$ hours after the experiment began.
The population decreases between $\boldsymbol{x<1}$ and $\boldsymbol{3