Question

consider the functions f and g in the tables below.
\\( f(x) = 90x^2 + 180x + 92 \\)

x	y
1	362
2	812
3	1,442
4	2,252
5	3,242

\\( g(x) = 6^x \\)

x	y
1	6
2	36
3	216
4	1,296
5	7,776

which of the following statements is true?
a. at approximately \\( x = 4.39 \\), the rate of change of \\( f \\) is equal to the rate of change of \\( g \\).
b. as \\( x \\) increases, the rate of change of \\( f \\) exceeds the rate of change of \\( g \\).
c. as \\( x \\) increases, the rate of change of \\( g \\) exceeds the rate of change of \\( f \\).
d. for every value of \\( x \\), the rate of change of \\( g \\) exceeds the rate of change of \\( f \\).

Explanation:

Step1: Analyze the functions

• \( f(x) = 90x^2 + 180x + 92 \) is a quadratic function (degree 2). Its rate of change (derivative) is \( f'(x)=180x + 180 \), which is a linear function (degree 1).
• \( g(x)=6^x \) is an exponential function. Its rate of change (derivative) is \( g'(x)=6^x\ln(6) \), which is also an exponential function (grows faster than linear as \( x \) increases).

Step2: Evaluate the rate of change behavior

• For quadratic \( f(x) \), the rate of change \( f'(x) \) is linear, so it grows at a constant rate (slope \( 180 \)).
• For exponential \( g(x) \), the rate of change \( g'(x) \) is exponential, so it grows at an increasing rate (faster than linear as \( x \) becomes large).

Step3: Check each option

• Option A: To find where \( f'(x)=g'(x) \), solve \( 180x + 180 = 6^x\ln(6) \). By numerical methods (e.g., graphing or iteration), we find that around \( x \approx 4.39 \), the two derivatives are equal. But let's check other options.
• Option B: As \( x \) increases, the exponential rate of change (for \( g \)) will eventually outpace the linear rate of change (for \( f \)), so \( f \)'s rate does not always exceed \( g \)'s.
• Option C: As \( x \) increases, the exponential function's derivative (rate of change) grows faster than the linear function's derivative. So for large enough \( x \), \( g \)'s rate exceeds \( f \)'s. This is correct.
• Option D: At \( x = 0 \), \( f'(0)=180(0)+180 = 180 \), \( g'(0)=6^0\ln(6)=\ln(6)\approx 1.79 \). So \( g'(0) < f'(0) \), so this is false.

Answer:

C. As \( x \) increases, the rate of change of \( g \) exceeds the rate of change of \( f \).