Question

For the function \[ f(x)=ax+\frac{b}{x}, \qquad a>0,\; b>0, \] which of the following statements are correct?  

(A) Function \(f(x)\) is increasing on \[ \left(\sqrt{\frac{b}{a}},\,\infty\right) \] 

(B) Function \(f(x)\) is increasing on \[ (-\infty,\infty) \] 

(C) Function \(f(x)\) is decreasing on \[ \left(-\sqrt{\frac{b}{a}},\,\sqrt{\frac{b}{a}}\right) \] 

(D) Function \(f(x)\) is increasing on \[ \left(-\infty,\,-\sqrt{\frac{b}{a}}\right) \]

• A. (A), (B) and (D) only
• B. (A) and (C) only
• C. (A), (C) and (D) only
• D. (A) and (D) only

Hint:

Always remember to exclude points where the function is undefined (like \( x=0 \) here) when determining intervals of increase and decrease.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Check where \( f'(x) > 0 \) (increasing) and \( f'(x) < 0 \) (decreasing).

Step 2: Detailed Explanation:
\( f'(x) = a - \frac{b}{x^2} = \frac{ax^2 - b}{x^2} \).
Critical points are \( x^2 = b/a \implies x = \pm \sqrt{b/a} \).
For \( |x| > \sqrt{b/a} \), \( f'(x) > 0 \) (increasing).
For \( |x| < \sqrt{b/a} \), \( f'(x) < 0 \) (decreasing).
Statement (A) and (D) are correct.

Step 3: Final Answer:
(A) and (D) are correct.