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:
A function is increasing where its derivative \( f'(x) > 0 \) and decreasing where \( f'(x) < 0 \).

Step 2: Detailed Explanation:
Calculate the derivative: \( f'(x) = a - \frac{b}{x^2} = \frac{ax^2 - b}{x^2} \).
Critical points are where \( f'(x) = 0 \), so \( ax^2 = b \implies x^2 = b/a \implies x = \pm \sqrt{b/a} \).
Let \( k = \sqrt{b/a} \). The derivative is \( f'(x) = \frac{a(x-k)(x+k)}{x^2} \).

• For \( x > k \) (i.e., \( x > \sqrt{b/a} \)): \( f'(x) > 0 \). The function is increasing. (Statement A is correct)
• For \( x < -k \) (i.e., \( x < -\sqrt{b/a} \)): \( f'(x) > 0 \). The function is increasing. (Statement D is correct)
• For \( -k < x < k \) (excluding \( x=0 \)): \( f'(x) < 0 \). The function is decreasing. (Statement C is partially descriptive but incorrect in range due to the discontinuity at \( x=0 \)).

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