Explanation Given: The function of the sales in thousands of dollars at the time t in months of the all seasonal items is approximated by, S ( t ) = − 2 t 3 + 33 t 2 − 108 t + 100 Here, t is the number corresponds to the month. If t = 1 , the month would be January, if t = 2 , the month would be February and so on. Formula used: Write the formula for derivative of power rule. If f ( x ) = x n for any number n , then the derivative. f ′ ( x ) = n x n − 1 If f ( x ) = k for any real number k , then the derivative is defined by the formula. f ′ ( x ) = 0 To find local maximum and minimum find the values of x from the equation f ′ ( x ) = 0 . Then find f ″ ( x ) and put the values of x. If f ″ ( x ) equals some positive quantity then it will be Local minimum is located at that point x and if f ″ ( x ) equals some negative quantity then Local maximum is located at the point at that point. Calculation: Recall the given expression. S ( t ) = − 2 t 3 + 33 t 2 − 108 t + 100 Differentiate with respect to t by using the power rule for derivative of the function. S ′ ( t ) = − 2 ( 3 t 3 − 1 ) + 33 ( 2 t 2 − 1 ) − 108 ( 1 ) + 0 = − 6 t 2 + 66 t − 108 …… (1) For the extreme point, calculate the critical point by equating S ′ ( t ) equal to zero. Substitute 0 for S ′ ( t ) . − 6 t 2 + 66 t − 108 = 0 Simplify the equation. t 2 − 11 t + 18 = 0 t 2 − 9 t − 2 t + 18 = 0 t ( t − 9 ) − 2 ( t − 9 ) = 0 ( t − 2 ) ( t − 9 ) = 0 Solve for t

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134767628

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 12.1, Problem 56E

To determine

To calculate: The function of the sales in thousands of dollars at the time t in months of the all seasonal items is approximated by S(t)=−2t3+33t2−108t+100. Here, t is the number corresponds to the month. If t=1, the month would be January, if t=2, the month would be February and so on.

The value of t and month at which the local minimum sales occur.

The value of t and month at which the local maximum sales occur.

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Chapter 12.1, Problem 59E

Chapter 12.1, Problem 57E

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Chapter 12 Solutions

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

Ch. 12.1 - Checkpoint 1 For what values of x is the function...Ch. 12.1 - Checkpoint 2 Find all intervals on which is...Ch. 12.1 - Checkpoint 3 Identity the x-values of all points...Ch. 12.1 - Checkpoint 4 Find the critical numbers for each of...Ch. 12.1 - Prob. 5CP Ch. 12.1 - Prob. 6CP Ch. 12.1 - Checkpoint 7 Find the locations of the local...Ch. 12.1 - Prob. 8CP Ch. 12.1 - Checkpoint 9 If a sales function is given by...Ch. 12.1 - Prob. 1E

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Solution Summary: The author calculates the function of sales in thousands of dollars at the time t in months of the all seasonal items.

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