Explanation Obtain the derivative of the functions f ( v ) = v ( a − v ) ( v − 1 ) as follows. f ′ ( v ) = ( v ( 0.25 − v ) ( v − 1 ) ) ′ = ( − v 3 + 1.25 v 2 − 0.25 ) ′ = − 3 v 2 + 2.5 v − 0.25 Set f ′ ( c ) = 0 and find the value of c . − 3 c 2 + 2.5 c − 0.25 = 0 c = − 2.5 ± ( 2.5 ) 2 + 4 ( − 3 ) 0.25 2 ( − 3 ) c = − 2.5 ± 3.25 2 ( − 3 ) c = − 2.5 ± 1.803 2 ( − 3 ) Simplify further as follows. c = − 2.5 ± 1.803 − 6 c = 2.5 − 1.803 6 and c = 2.5 + 1.803 6 c = 0.697 6 and c = 4.303 6 c ≈ 0.12 and c ≈ 0.72 Hence, the critical numbers are 0.12 and 0.72. These points determine the intervals ( 0 , 0.12 ) , ( 0.12 , 0.72 ) and ( 0.72 , 1 ) . Consider a point in each interval and substitute it in the derivative to find its sign. Consider v = 0.1 in the interval ( 0 , 0.12 ) and find the sign of the derivative at this point. f ′ ( 0.1 ) = − 3 ( 0.1 ) 2 + 2.5 ( 0.1 ) − 0.25 = − 0.03 + 0.25 − 0.25 = − 0.03 < 0 Consider v = 0.2 in the interval ( 0.12 , 0.72 ) and find the sign of the derivative at this point. f ′ ( 0.2 ) = − 3 ( 0.2 ) 2 + 2.5 ( 0.2 ) − 0

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Differential Equations

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Chapter 13, Problem 70RE

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To sketch: The graph of the function f(v)=v(a−v)(v−1) when a=0.25 within the range of 0≤v≤1 .

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Chapter 13, Problem 69RE

Chapter 13, Problem 71RE

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Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

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The graph of the function f ( v ) = v ( a − v ) ( v − 1 ) when a = 0.25 within the range of 0 ≤ v ≤ 1 .

Solution Summary: The author explains how to obtain the derivative of the functions f(a-v) and find the value of c.

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To sketch: The graph of the function f(v)=v(a−v)(v−1) when a=0.25 within the range of 0≤v≤1 .

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Chapter 13 Solutions