Alright, let's determine which statement about the rates of change of functions \( f(x) \) and \( g(x) \) is true.
We have two functions defined as:
[tex]\[ f(x) = 180x + 92 \][/tex]
[tex]\[ g(x) = 6^x \][/tex]
To compare their rates of change, we need to compute their derivatives. The rate of change of a function is given by its derivative.
The derivative of \( f(x) \):
[tex]\[ f'(x) = 180 \][/tex]
This is a constant rate of change since \( f(x) \) is a linear function. Therefore, \( f'(x) = 180 \) for all \( x \).
The derivative of \( g(x) \):
[tex]\[ g'(x) = 6^x \cdot \ln(6) \][/tex]
This is an exponential function whose rate of change increases as \( x \) increases.
To determine where the rates of change are equal, we need to solve for \( x \) where:
[tex]\[ f'(x) = g'(x) \][/tex]
[tex]\[ 180 = 6^x \cdot \ln(6) \][/tex]
Through solving, we have determined that this occurs when \( x \approx 2.57 \).
With this information, let's evaluate each statement:
A. At approximately \( x=4.39 \), the rate of change of \( f \) is equal to the rate of change of \( g \).
- We found that \( x \) where the rates of change are equal is approximately 2.57, not 4.39. So, this statement is false.
B. For every value of \( x \), the rate of change of \( g \) exceeds the rate of change of \( f \).
- This is not true because we have found a point \( x \approx 2.57 \) where their rates of change are equal. Also, for \( x \) less than 2.57, the rate of change of \( g \) is less than the rate of change of \( f \).
C. As \( x \) increases, the rate of change of \( g \) exceeds the rate of change of \( f \).
- This is true because \( g'(x) = 6^x \cdot \ln(6) \) grows exponentially and will exceed the constant rate of change \( f'(x) = 180 \) for values of \( x \) greater than 2.57.
D. As \( x \) increases, the rate of change of \( f \) exceeds the rate of change of \( g \).
- This is false. Although \( f \) has a constant rate of change, \( g' (x) \) increases exponentially, eventually surpassing and remaining greater than \( f' (x) \) for all \( x \) greater than 2.57.
Therefore, the correct statement is:
C. As [tex]\( x \)[/tex] increases, the rate of change of [tex]\( g \)[/tex] exceeds the rate of change of [tex]\( f \)[/tex].
