Question

if the population of krill in a piece of the ocean is modeled by p(t)=t^2 - 2t + 30, where t is the number of minutes after the initial population is measured, is there a moment in time between t = 2 minutes to t = 4 minutes where the population of krill is not changing? (that is, p(c)=0 for some 2 < c < 4.)

Explanation:

Step1: Find the derivative of \(P(t)\)

The derivative of \(P(t)=t^{2}-2t + 30\) using the power - rule \((x^{n})^\prime=nx^{n - 1}\) is \(P^\prime(t)=2t-2\).

Step2: Set \(P^\prime(t) = 0\) and solve for \(t\)

Set \(2t - 2=0\). Add 2 to both sides: \(2t=2\). Then divide by 2, we get \(t = 1\).

Step3: Check if there is a \(c\) in the interval \((2,4)\) such that \(P^\prime(c)=0\)

Since the solution of \(P^\prime(t)=0\) is \(t = 1\) and \(1
otin(2,4)\), there is no value \(c\) in the interval \((2,4)\) for which \(P^\prime(c)=0\). So the population of krill is changing for all \(t\in(2,4)\).

Answer:

No