If S_1 and S_2 are respectively the sets of local minimum and local maximum points of the function f(x) = 9x^4 + 12x^3 - 36x^2 + 25, x ∈ R, then:

To find the local minimum and maximum points, we need to find the critical points of the function f(x) = 9x^4 + 12x^3 - 36x^2 + 25. First, we find the first derivative:
f'(x) = 36x^3 + 36x^2 - 72x
Setting f'(x) = 0, we get:
36x^3 + 36x^2 - 72x = 0
36x(x^2 + x - 2) = 0
36x(x+2)(x-1) = 0
This gives us critical points at x = 0, x = -2, and x = 1.
Now, we find the second derivative:
f''(x) = 108x^2 + 72x - 72
Let's evaluate the second derivative at each critical point:
For x = -2: f''(-2) = 108(-2)^2 + 72(-2) - 72 = 432 - 144 - 72 = 216 > 0. Thus, x = -2 is a local minimum.
For x = 0: f''(0) = -72 < 0. Thus, x = 0 is a local maximum.
For x = 1: f''(1) = 108 + 72 - 72 = 108 > 0. Thus, x = 1 is a local minimum.
Therefore, the set of local minimum points S_1 = {-2, 1} and the set of local maximum points S_2 = {0}.
Correct option is D. S_1={-2,1};S_2={0}