Step 1:
Given the function f(x)=(x+3)^2(x-2)(x+1)+c, where c is a constant. We know that it has an absolute minimum point at (1, -32). This means when we substitute x = 1 into the function, the output should be -32. With this information, we can solve for the constant 'c'. Let's do that:
-32 = (1+3)^2(1-2)*(1+1) + c
-32 = 16*-1*2 + c
-32 = -32 + c
0 = c
Now, our function is f(x) = (x+3)^2(x-2)(x+1).
Step 2:
To find x-intercepts of the function, we need to solve for x when f(x) = 0. This means: (x+3)^2(x-2)(x+1) = 0.
Given the zero-product property, if ab = 0, then a = 0 or b = 0.
Therefore, the x-intercepts would be:
x+3 = 0 => x = -3,
x-2 = 0 => x = 2,
x+1 = 0 => x = -1.
For the function f(x) = (x+3)^2(x-2)(x+1), there are three distinct x-intercepts, so a) 3 x-intercepts corresponds to this function.
Step 3:
For the function to have 2 intercepts, it means 2 of our intercepts must coincide (be the same). This gives a repeated root.
This could happen in 2 cases:
Case 1: If the factor (x+3)^2 is the solution, then the x-intercepts would be x = -3 (twice), and x = 2.
Case 2: If the factor (x+1) combines with (x+3)^2 to be the solution (and they are the same), then the x-intercepts would be x = -1 (thrice) and x = 2.
So, b) 2 x-intercepts corresponds to these two possibilities.
Step 4:
For the function to have 0 intercepts, it would mean there are no real roots. The function doesn't cross the x-axis. However, because it's a polynomial function of degree 4, it must necessarily have 4 roots due to the Fundamental theorem of Algebra - but in the case, these roots could be complex (with imaginary parts).
In this case, adjusting the constant 'c' from zero to a different value is likely necessary, to shift it up or down, and this would require more information to define this properly.
In conclusion, for c), 0 x-intercepts are unlikely for this function with real roots, but it's theoretically possible with complex roots.
Note: This answer doesn't involve the local maximum point at (-1.75, 4.39); additional calculations (like taking derivatives) with this point could account for 'c' or other adjustments, but doesn't directly correlate with the question posed regarding x-intercepts.
Please consult with a mathematical software or geometry tool to graph these functions and further enhance your understanding.