Final answer:
To determine the extrema of the function f(x) = x⁴ - 2x² + x - 2, we find and analyze its critical points using the first and second derivatives. A positive second derivative at a critical point implies a local minimum, and a negative second derivative implies a local maximum. The function's quartic nature guarantees an absolute minimum and maximum.
Step-by-step explanation:
The function given is f(x) = x⁴ - 2x² + x - 2, and we are interested in finding its extrema. First, we take the derivative of the function to find the critical points. The derivative of f(x) is f'(x) = 4x³ - 4x + 1. We find critical points by setting f'(x) to zero and solving the equation 4x³ - 4x + 1 = 0 for x.
To determine the type of the critical points, we take the second derivative f''(x) = 12x² - 4 and evaluate it at each critical point. If f''(x) > 0 at a critical point, it's a local minimum; if f''(x) < 0, it's a local maximum; and if f''(x) = 0, further analysis is needed. Since the function is a polynomial and hence continuous and differentiable everywhere, if there is a local extrema, they could also be absolute extrema, depending on the behavior of the function as x approaches infinity.
Without solving for the actual points, given the information that the second derivative is negative at x = 0, we can deduce that this point is a local maximum. If there is a point where the second derivative is positive, that point is a local minimum. Remember, a function of this nature (quartic) will always have an absolute minimum and maximum as x becomes large negative or large positive due to the leading term x⁴.