Question

The interval in which the function \( f(x) = \frac{4x^2 + 1}{x} \) is decreasing is:

• A. \( \left( -\frac{1}{2}, \frac{1}{2} \right) \)
• B. \( \left[ -\frac{1}{2}, \frac{1}{2} \right] \)
• C. \( (-1, 1) \)
• D. \( [-1, 1] \)

Hint:

To determine where a function is increasing or decreasing, calculate its derivative and analyze its sign.

Answer 1

The Correct Option is A

Step 1: To find the interval where the function is decreasing, we need to compute the derivative of the function.
Step 2: The first derivative is \( f'(x) = \frac{8x}{x^2} - \frac{4}{x^2} \).
Step 3: Solve \( f'(x) = 0 \), and we find the critical points. The function is decreasing on \( \left( -\frac{1}{2}, \frac{1}{2} \right) \).

Final Answer: \[ \boxed{\left( -\frac{1}{2}, \frac{1}{2} \right)} \]