Answer $A$
Given: $$f(x) = x^3-6x^2+24x$$
First derivative of above equation is:
$$f'(x) = 3x^2-12x+24 = 3\underbrace{(x^2-4x+8)}_\text{D<0}$$
Since, $$a > 0, D = b^2 -4ac < 0 \implies$$ Parabola is upward facing with no real roots.
Now, Vertex of the parabola, $$x-coordinate = \frac{-b}{2a} = 2$$
Putting this value in above equation, we will get the y-coordinate:
$$y-coordinate = 3(2^2-4.2+8) = 12 $$
So, the coordinates of the vertex are $$(2, 12)$$
So, $(A)$ is the right option.