Consider the function f (x) = −x⁴ + 2x² + 3.

(a) Determine the maximum number of turning points on the graph of f.

(b) Graph f using a graphing utility with window settings [−5, 5, 1, −10, 10, 1]. Verify that the graph has the maximum number of turning points found in part (a).

(c) Determine the end behavior of f ; that is, find the power function that the graph of f resembles for large values of x .

(d) Based on the results of parts (b) and (c), explain why the graph of f will not have any additional turning points off the viewing window.

(e) The function f is decreasing where its derivative f'(x) = −4x³ + 4x ≤ 0. Use the derivative to determine the intervals for which f is decreasing. Because polynomials are continuous over their domain, all endpoints are included in the interval describing increasing/decreasing. However, in general, the numbers at the endpoints must be tested separately to determine if they should be included in the interval describing where a function is increasing or decreasing.

(f) Use a graphing utility to determine the intervals for which f is decreasing to confirm your results from part (e).