Final answer:
To find the extrema of the function, calculate the first derivative to find potential critical points, then use the second derivative test to determine whether those points are local minima or maxima. The function has no absolute maximum due to its leading term and will have an absolute minimum since it is bounded below.
Step-by-step explanation:
From the given function f(x) = x´ - 2x² + x + 2, we can identify the extrema by looking for points where the first derivative equals zero and then using the second derivative test to determine the nature of each extremum. This involves calculating f'(x) for the first derivative and f''(x) for the second derivative. For the given function:
f'(x) = 4x³ - 4x + 1
f''(x) = 12x² - 4
Setting f'(x) to zero and solving for x will give us the potential points for local extrema. Applying the second derivative test, if f''(x) is positive at a critical point, then the function has a local minimum there; if f''(x) is negative, then the function has a local maximum there. To determine if the function has absolute maxima or minima, we must consider the behavior of the function as x approaches infinity or negative infinity. Since the leading term is x´, which is always positive for large values of x and negative infinity, the function will tend towards positive infinity, indicating that there's no absolute maximum, but since the function will increase without bound, there must be an absolute minimum at some point.
We also need to consider the examples provided which hint towards finding local and absolute extrema. For example, the given information helped us understand that at x=3, f(x) has a positive value and a positive slope that is decreasing, suggesting a point where the function might begin to curve downwards, potentially near a local maximum or on the rising side of a local minimum.
To conclude, for the function f(x) = x´ - 2x² + x + 2, we can first find the critical points by setting the first derivative to zero, then apply the second derivative test to each critical point to determine whether they represent local maxima or minima. The absence of an upper bound for the function indicates the lack of an absolute maximum, and the extremal behavior as x approaches infinity or negative infinity helps establish the presence of an absolute minimum.