To determine To calculate: The water temperature that produces the maximum number of salmon swimming upstream. Explanation Given: The provided function: s ( x ) = − x 3 + 3 x 2 + 400 x + 5000 Where; x lies in the interval [ 6 , 20 ] and it represents the temperature of the water in degree Celsius. Formula used: The derivative property as; d d x ( x n ) = n x n − 1 And According to the first derivative test, if c is the only critical number of the given function for an interval [ a , b ] such that a < c < b . Consider, f is differential for every value of x in the interval except at x = c . 1. For f ′ ( a ) > 0 and f ′ ( b ) < 0 , there will be a local maxima at c . 2. For f ′ ( a ) < 0 and f ′ ( b ) > 0 , there will be a local minima at c . 3. There will be no local extrema when f ′ ( a ) and f ′ ( b ) are either both positive or both negative Calculation: Consider the provided function: s ( x ) = − x 3 + 3 x 2 + 400 x + 5000 To get the water temperature that produces the maximum number of saloon put s ′ ( x ) = 0 Use derivative formulae in the above function which is: d d x ( x n ) = n x n − 1 and d d x ( c ) = 0 Where; c constant value and its derivative is zero then, d d x ( − x 3 + 3 x 2 + 400 x + 5000 ) = 3 ( − x 2 ) + 6 x + 400 = − 3 x 2 + 6 x + 400 Now put s ′ ( x ) = 0 .That is, − 3 x 2 + 6 x + 400 = 0 3 x 2 − 6 x − 400 = 0 The above expression is quadratic equation so use quadratic formulae that are written below: x = − b ± b 2 − 4 ⋅ a ⋅ c 2 a The general form of a quadratic equation is as follows: f ( x ) = a x 2 + b x + c In which a , b , c are real number and a ≠ 0 Compare given quadratic function with the general form of quadratic function then the resulting value will be written below: a = 3 , b = − 6 , c = − 400 Then, x = − b ± b 2 − 4 ⋅ a ⋅ c 2 a = − ( − 6 ) ± ( − 6 ) 2 − 4 ⋅ 3 ⋅ ( − 400 ) 2 ⋅ 3 = 6 ± 36 + 4800 6 = 6 ± 4836 6 Further solve and get, x = 6 ± 4836 6 = 6 ± 69

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 59E

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To calculate: The water temperature that produces the maximum number of salmon swimming upstream.

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Chapter 12.1, Problem 55E

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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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