Explanation Given: The function is f ( x ) = x 5 3 − x . Formula used: Test for concavity: Let f ″ ( x ) exist for all x ∈ ( a , b ) , then · y = f ″ ( x ) > 0 for all x ∈ ( a , b ) , here the function f is concave upward on ( a , b ) . · y = f ″ ( x ) < 0 for all x ∈ ( a , b ) , here the function f is concave downward on ( a , b ) . y = f ( x ) has a point of inflection if f ″ ( x ) = 0 . To find the critical points substitute f ′ ( x ) = 0 . Second derivative test: If f ′ ( c ) = 0 and f ″ ( c ) exists, then f ( c ) is a local maximum if f ″ ( c ) < 0 . f ( c ) is a local minimum if f ″ ( c ) > 0 . The test fails if f ″ ( c ) = 0 . When test fails then use the first derivative test. Calculation: The function is f ( x ) = x 5 3 − x . First, find the derivative. f ′ ( x ) = 5 3 x 5 3 − 1 − 1 = 5 3 x 2 3 − 1 Now, find the second derivative. f ″ ( x ) = 5 3 ( 2 3 ) x 2 3 − 1 = 10 9 x − 1 3 Find the value of x by substitute f ″ ( x ) = 0 . 10 9 x − 1 3 = 0 x = 0 Thus, the zero of the f ″ ( x ) is x = 0 . This divides the x − axis into two intervals: ( − ∞ , 0 ) , and ( 0 , ∞ ) . Interval Point Function Sign of f ″ Behavior of f ( x ) ( − ∞ , 0 ) − 1 10 9 x − 1 3 = 10 9 ( − 1 ) − 1 3 = − 10 9 − v e Concave down ( 0 , ∞ ) 1 10 9 x − 1 3 = 10 9 ( 1 ) − 1 3 = 10 9 + v e Concave up Thus, f is concave up for interval ( 0 , ∞ ) and f is concave down for interval ( − ∞ , 0 )

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 4.4, Problem 47E

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Find the intervals on which f is concave up or down, the points of inflection, the critical points and the local minima and maxima.

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Chapter 4.4, Problem 46E

Chapter 4.4, Problem 48E

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Chapter 4 Solutions

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Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

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Find the intervals on which f is concave up or down, the points of inflection, the critical points and the local minima and maxima .

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Find the intervals on which f is concave up or down, the points of inflection, the critical points and the local minima and maxima.

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Chapter 4 Solutions