Question

The function \( f(x) = x^2 + \dfrac{54}{x} \)

• A. is increasing and has minimum value 27 in the interval \( (0, \infty) \)
• B. is decreasing and has neither maximum nor minimum in the interval \( (-\infty, 0) \)
• C. has maximum value 27 in the interval \( (-\infty, 0) \)
• D. is increasing and has neither maximum nor minimum values in the interval \( (-\infty, 0) \)

Hint:

Use the sign of the first derivative to determine increasing/decreasing behavior and extrema.

Answer 1

The Correct Option is B

Given \( f(x) = x^2 + \frac{54}{x} \) Find derivative: \[ f'(x) = 2x - \frac{54}{x^2} \] On the interval \( (-\infty, 0) \), both \( x < 0 \) and \( \frac{1}{x^2}>0 \), hence: \[ f'(x) = 2x - \frac{54}{x^2} < 0 \Rightarrow f(x) \text{ is decreasing} \] Since function keeps decreasing and does not attain a minimum or maximum on \( (-\infty, 0) \), the answer is as given.