The function \( f(x)\ =\ \displaystyle\frac{4x^2+1}{x} \) is decreasing in
(A). \( \left(\displaystyle\frac{-1}{2},\displaystyle\frac{1}{2}\right) \)
(B). \( \left(-\infty,\displaystyle\frac{-1}{2}\right) \)
(C). \( \left(0,\displaystyle\frac{1}{2}\right) \)
(D). \( \left(\displaystyle\frac{-1}{2},0\right) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (C) and (D) only
f is defined for x ≠ 0.
\( \displaystyle f(x)=4x+\frac{1}{x} \)
\( \displaystyle f'(x)=4-\frac{1}{x^2} \)
\( \displaystyle =\frac{4x^2-1}{x^2} \)
\( f'(x)\lt 0 \) when \( 4x^2\lt 1 \), i.e. \( -\frac{1}{2}\lt x\lt\frac{1}{2} \), x ≠ 0.
So f is decreasing on \( \left(-\frac{1}{2},0\right) \) and on \( \left(0,\frac{1}{2}\right) \), is not defined at x = 0, and is increasing for \( |x|\gt\frac{1}{2} \).
Checking the statements
(A) \( \left(\displaystyle\frac{-1}{2},\displaystyle\frac{1}{2}\right) \) — Incorrect. f is not defined at x = 0, which lies inside this interval.
(B) \( \left(-\infty,\displaystyle\frac{-1}{2}\right) \) — Incorrect. Here \( x^2\gt\frac{1}{4} \), so \( f'(x)\gt 0 \).
(C) \( \left(0,\displaystyle\frac{1}{2}\right) \) — Correct. Here \( x^2\lt\frac{1}{4} \), so \( f'(x)\lt 0 \).
(D) \( \left(\displaystyle\frac{-1}{2},0\right) \) — Correct. Here \( x^2\lt\frac{1}{4} \), so \( f'(x)\lt 0 \).