The interval in which the function \( f(x)=x^{4}-\displaystyle\frac{x^{3}}{3} \) is increasing, is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( (1/4,\infty) \)
A function is increasing where f′(x) > 0.
\( \displaystyle f'(x)=4x^{3}-x^{2} \)
\( \displaystyle =x^{2}(4x-1) \)
The factor \( x^{2} \) is positive for every \( x\ne0 \), so the sign of f′ is the sign of \( 4x-1 \).
\( \displaystyle 4x-1\gt0 \)
\( \displaystyle \Rightarrow\ x\gt\frac{1}{4} \)
So f is increasing in \( \left(\frac{1}{4},\infty\right) \).
Why the other options are incorrect
Option (1) \( (-\infty,1/4) \) — Incorrect. Here \( 4x-1\lt0 \), so f′(x) ≤ 0 and f is decreasing.
Option (2) \( (0,1/4) \) — Incorrect. Here \( 4x-1\lt0 \), so f is decreasing.
Option (4) \( (-\infty,-1/4) \) — Incorrect. Here \( 4x-1\lt0 \), so f is decreasing.