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The problem asks about a population changing according to the formula \( P(t) = 5.4^t \), and it provides three statements, asking which are always true.

We will go through each statement, analyze them, and then evaluate the answer choices.

### Step 1: Find the derivative of \( P(t) = 5.4^t \).

To find \( P'(t) \), we use the general formula for the derivative of an exponential function \( a^t \):
\[
\frac{d}{dt} a^t = a^t \ln(a)
\]
Here, \( a = 5.4 \), so
\[
P'(t) = 5.4^t \ln(5.4)
\]

### Step 2: Analyze the statements

#### Statement (i): 
\( P'(1) \) is less than the average rate of change of \( P \) between 0 and 1.

- \( P'(1) \) is the instantaneous rate of change at \( t = 1 \):
  \[
  P'(1) = 5.4^1 \ln(5.4) = 5.4 \ln(5.4)
  \]

- The average rate of change between \( t = 0 \) and \( t = 1 \) is:
  \[
  \text{Average Rate of Change} = \frac{P(1) - P(0)}{1 - 0} = P(1) - P(0) = 5.4^1 - 5.4^0 = 5.4 - 1 = 4.4
  \]

Now, compare \( P'(1) \) and the average rate of change:
- \( P'(1) = 5.4 \ln(5.4) \approx 5.4 \times 1.686 = 9.1144 \)
- The average rate of change = 4.4

So, \( P'(1) \) is greater than 4.4, making **statement (i) false**.

#### Statement (ii): The average rate of change between 0 and 1 is 5.4.

From the calculation above, the average rate of change between \( t = 0 \) and \( t = 1 \) is 4.4, not 5.4, so **statement (ii) is false**.

#### Statement (iii): The average rate of change between 0 and 1 is 5.4.

This is essentially the same as statement (ii), which we already determined to be false.

### Step 3: Conclusion
All statements (i), (ii), and (iii) are false. Therefore, the correct answer is **(D) none of them**.

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Would you like more details on this or have any questions? Here are 5 related questions to further explore:

1. What is the difference between instantaneous rate of change and average rate of change?
2. How do you derive the formula for the derivative of an exponential function?
3. Can the rate of change of a population ever be negative in this model?
4. What are the implications of an exponential growth model like \( P(t) = 5.4^t \) for real-world populations?
5. How would the conclusions change if \( P(t) \) had a different base, such as \( 2^t \)?

**Tip:** When comparing rates of change, always remember to check whether you are dealing with instantaneous rates (derivatives) or average rates (over an interval).

A population changes according to the formula P(t) = 5.4^t. Which of the following statements are always true? (i) P'(1) is less than the average rate of change of P between 0 and 1. (ii) The average rate of change between 0 and 1 is 5.4. (iii) The average rate of change between 0 and 1 is 5.4. The choices are: (A) (i) only, (B) (i) and (ii) only, (C) (i) and (iii) only, (D) none of them, (E) (ii) and (iii) only, (F) all of them, (G) (iii) only, (H) (ii) only.

Explore the rate of change of a population model defined by P(t) = 5.4^t. Solve for the derivative, analyze the average rate of change between intervals, and determine which of the given statements about the rates are always true. Suitable for advanced high school or early college calculus students.

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