To determine whether a function has a local maximum or minimum at a specific point, we need to analyze the behavior of the function around that point. In this case, we are given the function f(x) = x^5 - 5x^4 - 5x^3 + 12.

1. Analyzing the function:
Let's first find the derivative of f(x) to determine the critical points where the function may have a local maximum or minimum.
f'(x) = 5x^4 - 20x^3 - 15x^2

2. Finding the critical points:
To find the critical points, we set the derivative equal to zero and solve for x:
5x^4 - 20x^3 - 15x^2 = 0
Factoring out common terms:
x^2(5x^2 - 20x - 15) = 0

Setting each factor equal to zero:
x^2 = 0 or 5x^2 - 20x - 15 = 0

Solving the first equation, we find x = 0 as a critical point.

3. Analyzing the second equation:
To analyze the quadratic equation 5x^2 - 20x - 15 = 0, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)

Plugging in the values a = 5, b = -20, and c = -15, we get:
x = (-(-20) ± √((-20)^2 - 4(5)(-15)))/(2(5))
x = (20 ± √(400 + 300))/(10)
x = (20 ± √700)/10
x = (20 ± 10√7)/10
x = 2 ± √7

So, the other two critical points are x = 2 + √7 and x = 2 - √7.

4. Evaluating the function at the critical points:
Now, we need to evaluate the function f(x) at the critical points to determine whether they correspond to local maximum or minimum.

f(0) = (0)^5 - 5(0)^4 - 5(0)^3 + 12 = 12
f(2 + √7) = (2 + √7)^5 - 5(2 + √7)^4 - 5(2 + √7)^3 + 12
f(2 - √7) = (2 - √7)^5 - 5(2 - √7)^4 - 5(2 - √7)^3 + 12

5. Conclusion:
Based on our analysis, f(x) = x^5 - 5x^4 - 5x^3 + 12 has a critical point at x = 0. Evaluating the function at this point, we find that f(0) = 12. Since the function does not change sign around this point, it does not have a local maximum or minimum at x = 0.

Similarly, evaluating the function at x = 2 + √7 and x = 2 - √7, we find that the function does not have a local maximum or minimum at these points either.

Therefore, the points