20 1 Suppose F X 2 2 Cos Z For Z In 0 27 5 A Find All Critical Numbers Of F And Determine The Intervals Wher 1 (47.13 KiB) Viewed 72 times
20 1 Suppose F X 2 2 Cos Z For Z In 0 27 5 A Find All Critical Numbers Of F And Determine The Intervals Wher 2 (43.58 KiB) Viewed 72 times
[20] 1) Suppose f(x)=2+2 cos(z) for z in [0, 27). [5] a) Find all critical numbers of f and determine the intervals where f is increasing and the intervals where f is decreasing using sign analysis of f'. f'(x) = 1 + 2 (-siny) 1-2sin (x) = 0 ▼ Critical Numbers of f in (0,2m]:= ST Sign Analysis of f' (Number Line): t Intervals where f is increasing: [o, aut , 211] Intervals where f is decreasing: [] [2] b) Find all points where f has local extrema on [0,27] and use the First Derivative Test (from Section 3.3) to classify each local extrema as a local maximum or local minimum. Local Maxima (Points):__ F"(x) > 0 1-25mmv20 Local Minima (Points): I -√2 [2] c) Using the Closed Interval Method (from Section 3.1), find all points where f has absolute maximum and minimum values on (0, 2). Absolute Maxima (Points): 2 TI Absolute Minima (Points): 1 - 13 at y = at y = 2n1 1
[6] d) Using the partition numbers and sign analysis of f", find the intervals where f is concave upward and where f is concave downward. Find the inflection points of f. (1-25mm²x) = -2c0sx f"(z) = Partition Numbers of f" in [0, 2]: Sign Analysis for f" (Number Line): Intervals where f is concave upward:LX0 = 10x4 40 (3, 4) Intervals where f is concave downward: 1'Cosy.co (as 430 Inflection Points of f: x = 3 [5] e) Sketch the graph of y = f(x). Label the axes and indicate the scale on the axes. Label each local extrema (use "max" or "min") and inflection point (use "IP"). Suggestions: For the z-scale, divide [0, 2x] into 12 subintervals of equal length of /6. Determine the y-scale based on the absolute maximum and minimum of f found in part (c). 2
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