Maximum slope of the curve y = -x³ + 3x² + 9x – 27 is

The correct option is 0.

To understand why the correct answer is 0, we analyze the slope of the tangent to the given curve. The equation of the curve is:
$$y=-x^3+3x^2+9x-27$$

The slope of the curve at any point $x$ is given by its first derivative with respect to $x$, denoted as $\frac{dy}{dx}$:
$$\frac{dy}{dx}=\frac{d}{dx}\left(-x^3+3x^2+9x-27\right)$$
Using the power rule of differentiation, we get:
$$\frac{dy}{dx}=-3x^2+6x+9$$

To find the slope at the stationary points (turning points) of the curve, we set the first derivative (the slope) equal to 0:
$$-3x^2+6x+9=0$$

We can solve this quadratic equation by dividing the entire equation by $-3$:
$$x^2-2x-3=0$$
Factoring the quadratic equation:
$$\left(x-3\right)\left(x+1\right)=0$$
This gives the critical points:
$$x=3\text{ or }x=-1$$

At these stationary points, the tangent to the curve is horizontal, meaning the slope of the curve is exactly 0. Thus, the slope at these critical turning points of the curve is 0, which corresponds to the correct option.