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:

To check if a function is increasing or decreasing, use the derivative test: $f'(x) > 0$ means increasing, $f'(x) < 0$ means decreasing, and if $f'(x)$ changes sign, the function is neither increasing nor decreasing.

---

For $f(x) = 3x + 12$:

$f'(x) = 3$

Since $f'(x) = 3 > 0$ for all $x$, the function is increasing on R.

Statement A is correct.

---

For $f(x) = e^{2x}$:

$f'(x) = 2e^{2x}$

Since $e^{2x} > 0$ for all $x$, we have $f'(x) = 2e^{2x} > 0$ for all $x$.

The function is increasing on R, not decreasing.

Statement B is incorrect.

---

For $f(x) = x^2 - x - 1$ on interval $(-1, 1)$:

$f'(x) = 2x - 1$

Setting $f'(x) = 0$:

$2x - 1 = 0$

$x = \dfrac{1}{2}$

This critical point lies inside the interval $(-1, 1)$.

For $x = 0$ (left of $\dfrac{1}{2}$): $f'(0) = -1 < 0$, so the function is decreasing.

For $x = 0.8$ (right of $\dfrac{1}{2}$): $f'(0.8) = 0.6 > 0$, so the function is increasing.

The function decreases then increases on $(-1, 1)$, so it is neither strictly increasing nor strictly decreasing.

Statement C is correct.

---

For $f(x) = x^3 - 3x^2 + 4x$:

$f'(x) = 3x^2 - 6x + 4$

Completing the square:

$f'(x) = 3(x^2 - 2x) + 4$

$f'(x) = 3(x^2 - 2x + 1 - 1) + 4$

$f'(x) = 3(x - 1)^2 - 3 + 4$

$f'(x) = 3(x - 1)^2 + 1$

Since $3(x - 1)^2 \geq 0$ for all $x$, we have $f'(x) \geq 1 > 0$ for all $x$.

The function is increasing on R.

Statement D is correct.

---

Statements (A), (C), and (D) are correct.