For the function \( f(x)=ax+\displaystyle\frac{b}{x},\ a\gt 0,b\gt 0 \), which of the following statements are correct ?
(A) Function \( f(x) \) is increasing on \( \left(\sqrt{\displaystyle\frac{b}{a}},\infty\right) \)
(B) Function \( f(x) \) is increasing on \( (-\infty,\infty) \).
(C) Function \( f(x) \) is decreasing on \( \left(-\sqrt{\displaystyle\frac{b}{a}},\sqrt{\displaystyle\frac{b}{a}}\right) \)
(D) Function \( f(x) \) is increasing on \( \left(-\infty,-\sqrt{\displaystyle\frac{b}{a}}\right) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (D) only
\( \displaystyle f'(x)=a-\frac{b}{x^2} \)
\( \displaystyle =\frac{ax^2-b}{x^2} \)
\( f'(x)\gt 0 \) when \( x^2\gt\frac{b}{a} \), and \( f'(x)\lt 0 \) when \( 0\lt x^2\lt\frac{b}{a} \). Also, f is not defined at x = 0.
So f is increasing on \( \left(-\infty,-\sqrt{b/a}\right) \) and on \( \left(\sqrt{b/a},\infty\right) \), and decreasing on \( \left(-\sqrt{b/a},0\right) \) and on \( \left(0,\sqrt{b/a}\right) \) separately.
Checking the statements
(A) Function \( f(x) \) is increasing on \( \left(\sqrt{\displaystyle\frac{b}{a}},\infty\right) \) — Correct. Here \( x^2\gt\frac{b}{a} \), so \( f'(x)\gt 0 \).
(B) Function \( f(x) \) is increasing on \( (-\infty,\infty) \). — Incorrect. f is not defined at x = 0, and \( f'(x)\lt 0 \) for \( 0\lt x^2\lt\frac{b}{a} \).
(C) Function \( f(x) \) is decreasing on \( \left(-\sqrt{\displaystyle\frac{b}{a}},\sqrt{\displaystyle\frac{b}{a}}\right) \) — Incorrect. This interval contains x = 0, where f is not defined; f decreases only on each half separately.
(D) Function \( f(x) \) is increasing on \( \left(-\infty,-\sqrt{\displaystyle\frac{b}{a}}\right) \) — Correct. Here also \( x^2\gt\frac{b}{a} \), so \( f'(x)\gt 0 \).