The largest interval in which the function \(f(x)=x^3-3x^2+4x+1\), \(x\in R\) is increasing, is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → R
\(f'(x) = 3x^2-6x+4\)
Completing the square:
\(f'(x) = 3(x^2-2x+1)+1\)
\(= 3(x-1)^2+1\)
Since \((x-1)^2 \ge 0\), we get \(f'(x) \ge 1 \gt 0\) for every real x.
So f is (strictly) increasing on the whole real line, and the largest interval of increase is R. The intervals in Options (1), (2) and (3) are only parts of R.
NCERT (2026-27 edition):
Increasing and Decreasing Functions
