EXAMPLE 6 Discuss the curve y = 3x4 - 48x3 with respect to concavity, points of inflection, and local maxima and minima. Use this information to sketch the curve. SOLUTION If f(x) = 3x4 – 48x³, then f'(x) = 12x3 - 144x² = 12x?(x – 12) f"(x) = 36x2 – 288x = 36x(x – 8). 12 To find the critical numbers we set f'(x) = 0 and obtain x = 0 and x = . To use the Second Derivative Test we evaluate f" at these critical numbers: f"(0) = Video Example ) f"(12) = and f"(12) > 0, f(12) = is a local minimum. Since f"(0) = , the Second Since f'(12) = Derivative Test gives no information about the critical number 0. But since f'(x) < 0 for x <0 and also for 0 < x < 12, the First Derivative Test tells us that f does not have a local maximum or minimum at 0. [In fact, the expression for f'(x) shows that f decreases to the left of 12 and increase to the right of 12.] Since f"(x) = 0 when x = 0 or x = , we divide the real number line into intervals with these numbers as endpoints and complete the following chart. Interval f"(x) = 36x(x - 8) Concavity (-00, 0) upward (0.O) downward upward The point (0, 0) is an inflection point since the curve changes from concave upward to concave downward there. Also, -12288) is an inflection point since the curve changes from concave downward to concave upward there. Using the local minimum, the intervals of concavity, and the inflection points, we sketch the curve in the figure.
EXAMPLE 6 Discuss the curve y = 3x4 - 48x3 with respect to concavity, points of inflection, and local maxima and minima. Use this information to sketch the curve. SOLUTION If f(x) = 3x4 – 48x³, then f'(x) = 12x3 - 144x² = 12x?(x – 12) f"(x) = 36x2 – 288x = 36x(x – 8). 12 To find the critical numbers we set f'(x) = 0 and obtain x = 0 and x = . To use the Second Derivative Test we evaluate f" at these critical numbers: f"(0) = Video Example ) f"(12) = and f"(12) > 0, f(12) = is a local minimum. Since f"(0) = , the Second Since f'(12) = Derivative Test gives no information about the critical number 0. But since f'(x) < 0 for x <0 and also for 0 < x < 12, the First Derivative Test tells us that f does not have a local maximum or minimum at 0. [In fact, the expression for f'(x) shows that f decreases to the left of 12 and increase to the right of 12.] Since f"(x) = 0 when x = 0 or x = , we divide the real number line into intervals with these numbers as endpoints and complete the following chart. Interval f"(x) = 36x(x - 8) Concavity (-00, 0) upward (0.O) downward upward The point (0, 0) is an inflection point since the curve changes from concave upward to concave downward there. Also, -12288) is an inflection point since the curve changes from concave downward to concave upward there. Using the local minimum, the intervals of concavity, and the inflection points, we sketch the curve in the figure.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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