Question

Consider the equation below. (if an answer does not exist, enter dne.) f(x) = x/(x ^ 2 9) (a) find the interval on which fis increasing. (enter your answer using interval notation.)

Answer 1

To find the interval on which f(x) is increasing, we need to determine where the derivative of f(x) is positive. The interval on which f(x) is increasing is (-∞, -3) U (3, ∞).

Step-by-step explanation:

To find the interval on which f(x) is increasing, we need to determine where the derivative of f(x) is positive. The derivative of f(x) is found by using the quotient rule:

f'(x) = (1*(x^2 + 9) - x*(2x))/(x^2 + 9)^2

Simplifying this expression gives:

f'(x) = (9 - x^2)/(x^2 + 9)^2

To find where f'(x) is positive, we need to solve the inequality:

(9 - x^2)/(x^2 + 9)^2 > 0

Using a sign chart, we can determine that f'(x) is positive for x < -3 and x > 3. Therefore, the interval on which f(x) is increasing is (-∞, -3) U (3, ∞).