Which of the following are correct?
(A) The function \(f(x)=3x+12\) is increasing on R.
(B) The function \(f(x) = e^{2x}\) is decreasing on R.
(C) The function \(f(x) = x^2-x+1\) is neither increasing nor decreasing on \((-1, 1)\).
(D) The function \(f(x) = x^3-3x^2+4x\) is increasing on R.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
A function is increasing where its derivative is positive and decreasing where its derivative is negative.
(A) The function \(f(x)=3x+12\) is increasing on R. — Correct. \(f'(x) = 3 \gt 0\) for every x.
(B) The function \(f(x) = e^{2x}\) is decreasing on R. — Incorrect. \(f'(x) = 2e^{2x} \gt 0\) for every x, so f is increasing on R, not decreasing.
(C) The function \(f(x) = x^2-x+1\) is neither increasing nor decreasing on \((-1, 1)\). — Correct. The derivative changes sign at \(x = \frac{1}{2}\):
\(f'(x) = 2x - 1\)
\( \displaystyle f'(x) \lt 0 \text{ on } \left(-1, \frac{1}{2}\right), \qquad f'(x) \gt 0 \text{ on } \left(\frac{1}{2}, 1\right) \)
f first decreases and then increases on \((-1, 1)\), so it is neither increasing nor decreasing on the whole interval.
(D) The function \(f(x) = x^3-3x^2+4x\) is increasing on R. — Correct. Completing the square in the derivative:
\(f'(x) = 3x^2 - 6x + 4 = 3(x - 1)^2 + 1\)
So \(f'(x) \ge 1 \gt 0\) for every x, and f is increasing on R.