Question

F(x)=4eˣ + 9 g(x)=3eˣ + 8 use l'hospital's rule to compare the growth rate of f(x) and g(x).

a. f(x) grows faster.
b. g(x) grows faster.
c. they grow at the same rate.
d. there is not enough information.

Answer 1

To compare the growth rates of f(x) and g(x), we examine their derivatives. Since the derivative of f(x) is constantly greater than that of g(x), f(x) grows faster, indicating exponential growth.

Step-by-step explanation:

In order to compare the growth rates of the functions f(x) and g(x) given by f(x) = 4ex + 9 and g(x) = 3ex + 8, respectively, we can analyze their derivatives since the derivative represents the instantaneous rate of change, or growth rate, of a function. The exponential function ex has the unique property that its derivative is itself, ex. Hence, the derivatives of f(x) and g(x) with respect to x are f'(x) = 4ex and g'(x) = 3ex.

Comparing the derivatives, since ex is always positive for all x, and 4 is greater than 3, it follows that f'(x) > g'(x) for all x. Therefore, f(x) grows faster than g(x) at any point x. This is a characteristic of exponential growth where the growth rate is proportional to the current value. Thus, the correct answer is (a) f(x) grows faster.