Question

Select the correct answer.
Consider the functions f and g in the tables below.

f(x) = 90x2 + 180x + 92
x y
0 92
1 362
2 812
3 1,442
4 2,252
5 3,242
g(x) = 6x
x y
0 1
1 6
2 36
3 216
4 1,296
5 7,776

Which of the following statements is true?

A.
As x increases, the rate of change of f exceeds the rate of change of g.
B.
As x increases, the rate of change of g exceeds the rate of change of f.
C.
At approximately x = 4.39, the rate of change of f is equal to the rate of change of g.
D.
For every value of x, the rate of change of g exceeds the rate of change of f.

Answer 1

To evaluate the statements, we need to consider the rate of change of each function.

1. For **f(x) = 90x² + 180x + 92**, the derivative is:
\[
f'(x) = 180x + 180
\]
This indicates that the rate of change of f is linear and increases as x increases.

2. For **g(x) = 6x**, the derivative is:
\[
g'(x) = 6
\]
The rate of change of g is constant (6) and does not change with x.

Next, let's compare the rate of change of f and g as x increases:

- For **x = 0**:
- f'(0) = 180(0) + 180 = **180**
- g'(0) = **6**

Here, the rate of change of f exceeds that of g.

- For **x = 1**:
- f'(1) = 180(1) + 180 = **360**
- g'(1) = **6**

Again, the rate of change of f exceeds that of g.

- For **x = 2**:
- f'(2) = 180(2) + 180 = **540**
- g'(2) = **6**

Still, f's rate exceeds g's.

- For larger values of x:
- f'(3) = 180(3) + 180 = **720**
- g'(3) = **6**

- f'(4) = 180(4) + 180 = **900**
- g'(4) = **6**

It is clear from the pattern that as x increases, the rate of change of f continues to exceed that of g because of the quadratic nature of f, which gives it an increasing derivative.

Given this information, we can analyze the provided statements:
- **A**: True. As x increases, the rate of change of f (which increases linearly) exceeds the rate of change of g (which remains constant at 6).
- **B**: False. The rate of change of f exceeds that of g.
- **C**: This statement suggests an equality exists at some point. Given that the rate of change for f keeps increasing without bounds while g remains constant, this is likely false.
- **D**: False. The rate of change of g does not exceed that of f.

Thus, the true statement is:

**A. As x increases, the rate of change of f exceeds the rate of change of g.**