Final answer:
The domain of the function f(x) is all real numbers. Critical points, extremum values, and inflection points can be found by analyzing the derivative and second derivative of the function.
Step-by-step explanation:
To find the domain of the function f(x) = 2(x + 1)^3(x^2 - 1)(x - 1), we need to consider any restrictions on x that would make the function undefined. In this case, there are no restrictions, so the domain is all real numbers.
To find the critical points of the function, we need to find where the derivative of f(x) is equal to zero or undefined. The derivative is f'(x) = 6(x+1)^2(x^2-1)(x-1) + 2(x+1)^3(2x)(x-1) + 2(x+1)^3(x^2-1), which simplifies to f'(x) = 18(x+1)^2(x^2-1)(x-1) + 4x(x+1)^3(x-1).
To find the extremum values, we need to determine the sign of f'(x) around the critical points. If f'(x) changes sign from positive to negative, there is a local maximum. If it changes sign from negative to positive, there is a local minimum. The inflection points occur where the second derivative f''(x) changes sign.
Vertical asymptotes occur when the function approaches positive or negative infinity as x approaches a certain value. Horizontal asymptotes occur when the function approaches a constant value as x approaches positive or negative infinity. Slant asymptotes occur when the function approaches a linear function as x approaches positive or negative infinity.
To determine intervals of increasing, decreasing, concave up, and concave down, we need to examine the sign of f'(x) and f''(x) within given intervals.
Limits are used to understand the behavior of a function as it approaches specific values or as x tends to positive or negative infinity. In this context, limits help us understand the behavior of f(x) as x approaches certain points or as x goes to infinity, and they provide information about the function's asymptotes and the existence of extremum values or inflection points.