Question

A coordinate plane shows an increasing solid curve and a dotted straight line plotted on it.
The graph of f(x)=100⋅1.5x
and g(x)=100x+100
is shown. Which of the following statements is true?

Statement #1: While the growth rate of f(x)
is initially greater than the growth rate of g(x)
, the growth rate of g(x)
keeps increasing and eventually surpasses the growth rate of f(x)
.

Statement #2: The growth rate of g(x)
is greater than the growth rate of f(x)
between approximately x=0
and x=4
.

Statement #3: While the growth rate of g(x)
is initially greater than the growth rate of f(x)
, the growth rate of f(x)
keeps increasing and, by x=2.5
, surpasses the growth rate of g(x)
.

(1 point)
Statement #
is true.

Answer 1

To analyze the given functions \( f(x) = 100 \cdot 1.5^x \) and \( g(x) = 100x + 100 \), we need to understand their growth rates.

1. **For \( f(x) = 100 \cdot 1.5^x \)**:
- This is an exponential function. The growth rate of exponential functions increases as \( x \) increases.
- Specifically, the derivative \( f'(x) \) can be computed as follows:
\[
f'(x) = 100 \cdot 1.5^x \cdot \ln(1.5)
\]
- Since \( 1.5 \) is greater than \( 1 \), \( f'(x) \) is always positive and increasing.

2. **For \( g(x) = 100x + 100 \)**:
- This is a linear function. The growth rate of linear functions remains constant. The derivative is:
\[
g'(x) = 100
\]
- So the growth rate of \( g(x) \) remains \( 100 \) for all \( x \).

### Evaluating the Statements:

- **Statement #1**: While the growth rate of \( f(x) \) is initially greater than the growth rate of \( g(x) \), the growth rate of \( g(x) \) keeps increasing and eventually surpasses the growth rate of \( f(x) \).
- **False**: The growth rate of \( g(x) \) is constant, while \( f(x) \) has an increasing growth rate.

- **Statement #2**: The growth rate of \( g(x) \) is greater than the growth rate of \( f(x) \) between approximately \( x=0 \) and \( x=4 \).
- To determine this, evaluate the growth rates:
- At \( x=0 \):
\[
f'(0) = 100 \cdot 1.5^0 \cdot \ln(1.5) \approx 100 \cdot 0.405 = 40.5
\]
- Thus, \( g'(0) = 100 > 40.5 \).
- At \( x=4 \):
\[
f'(4) = 100 \cdot 1.5^4 \cdot \ln(1.5) \approx 100 \cdot 5.0625 \cdot 0.405 \approx 204.2
\]
- \( g'(4) = 100 \) remains the same.
- So, \( g(x) \)'s growth rate is greater than \( f(x) \) at least for some values between \( 0 \) and \( 4 \). Thus this statement is **partially true** but suggesting it's true over the entire interval.

- **Statement #3**: While the growth rate of \( g(x) \) is initially greater than the growth rate of \( f(x) \), the growth rate of \( f(x) \) keeps increasing and, by \( x=2.5 \), surpasses the growth rate of \( g(x) \).
- Evaluate at \( x=2.5 \):
\[
f'(2.5) = 100 \cdot 1.5^{2.5} \ln(1.5) \approx 100 \cdot 2.838 \cdot 0.405 \approx 114.5
\]
- Therefore, \( f'(2.5) > g'(2.5) = 100 \).
- This statement is **true**.

Based on this analysis, **Statement #3** is the true statement.