Which of the following functions are strictly increasing on R?
(A) \(f(x) = e^{-x}\)
(B) \(f(x) = x^{3} + 2x\)
(C) \(f(x) = x^{2} + 1\)
(D) \(f(x) = 2x + \sin x\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B) and (D) only
A function is strictly increasing on R when \(f'(x) \gt 0\) for every real x. The derivatives are
(A) \( f'(x) = -e^{-x} \)
(B) \( f'(x) = 3x^{2} + 2 \)
(C) \( f'(x) = 2x \)
(D) \( f'(x) = 2 + \cos x \)
So the derivatives in (B) and (D) are positive for all x, the one in (A) is negative for all x, and the one in (C) changes sign at x = 0.
Checking the statements
(A) \(f(x) = e^{-x}\) — Incorrect. \(f'(x) = -e^{-x} \lt 0\), so it is strictly decreasing on R.
(B) \(f(x) = x^{3} + 2x\) — Correct. \(f'(x) = 3x^{2} + 2 \ge 2 \gt 0\).
(C) \(f(x) = x^{2} + 1\) — Incorrect. \(f'(x) = 2x \lt 0\) for x < 0, so it decreases on \((-\infty, 0)\).
(D) \(f(x) = 2x + \sin x\) — Correct. \(f'(x) = 2 + \cos x \ge 2 - 1 = 1 \gt 0\).