For the function \(f(x) = x^3 + 3x + 2\), \(x \in R\):
(A) \(f'(x) = 3x^2 + 3\)
(B) f has no critical point
(C) f is strictly increasing on R
(D) f has a local minimum at \(x = 0\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (C) only
Differentiating by the power rule:
\(f'(x) = 3x^2 + 3\)
\(f'(x) = 3(x^2 + 1)\)
Since \(x^2 + 1 \ge 1\) for every real x, \(f'(x) \ge 3 \gt 0\) everywhere.
So \(f'(x) = 3x^2 + 3\) is never zero and always positive.
Checking the statements
(A) \(f'(x) = 3x^2 + 3\) — Correct. Power rule on \(x^3 + 3x + 2\).
(B) f has no critical point — Correct. \(3x^2 + 3 = 0\) has no real root.
(C) f is strictly increasing on R — Correct. \(f'(x) \ge 3 \gt 0\) for all x.
(D) f has a local minimum at \(x = 0\) — Incorrect. \(f'(0) = 3 \ne 0\), and f keeps increasing through x = 0.