Consider the following statements:
(A) \(f(x) = x^3 + 2x\) is strictly increasing on R.
(B) \(f(x) = e^{-x}\) is strictly increasing on R.
(C) \(f(x) = x^2\) is strictly decreasing on \((-\infty, 0)\).
(D) \(f(x) = \log x\) is strictly increasing on \((0, \infty)\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
A function is strictly increasing on an interval where its derivative is positive, and strictly decreasing where its derivative is negative. The derivatives are:
\(\frac{d}{dx}(x^3 + 2x) = 3x^2 + 2\)
\(\frac{d}{dx}(e^{-x}) = -e^{-x}\)
\(\frac{d}{dx}(x^2) = 2x\)
\(\frac{d}{dx}(\log x) = \frac{1}{x}\)
So \(3x^2 + 2 \gt 0\) for all x, \(-e^{-x} \lt 0\) for all x, \(2x \lt 0\) for x < 0, and \(\frac{1}{x} \gt 0\) for x > 0.
Checking the statements
(A) \(f(x) = x^3 + 2x\) is strictly increasing on R. — Correct. \(f'(x) = 3x^2 + 2 \ge 2 \gt 0\) for every x.
(B) \(f(x) = e^{-x}\) is strictly increasing on R. — Incorrect. \(f'(x) = -e^{-x} \lt 0\), so it is strictly decreasing on R.
(C) \(f(x) = x^2\) is strictly decreasing on \((-\infty, 0)\). — Correct. \(f'(x) = 2x \lt 0\) for every x < 0.
(D) \(f(x) = \log x\) is strictly increasing on \((0, \infty)\). — Correct. \(f'(x) = \frac{1}{x} \gt 0\) for every x > 0.