Explanation Given: The revenue function R ( x ) , that gives the revenue in thousands of dollars where x 1 represents the number in thousands of running shoes and x 2 represents the number of basketball shoes. R ( x ) = − 5 x 1 2 − 8 x 2 2 − 2 x 1 x 2 + 42 x 1 + 102 x 2 Formula used: The function has a relative minima at critical point ( a , b ) if f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 > 0 and f x x ( a , b ) > 0 . The function has a relative maxima at critical point ( a , b ) f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 > 0 and f x x ( a , b ) < 0 . A point ( a , b , f ( a , b ) ) is said to be a saddle point if f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 < 0 . The test fails when f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 = 0 Calculation: Equate both the partial derivatives of the revenue function to 0 to obtain the critical points. R x 1 ( x 1 , x 2 ) = − 10 x 1 − 2 x 2 + 42 − 10 x 1 − 2 x 2 + 42 = 0 x 1 = 42 − 2 x 2 10 And, R x 2 ( x 1 , x 2 ) = − 16 x 2 − 2 x 1 + 102 − 16 x 2 − 2 x 1 + 102 = 0 x 1 = 102 − 16 x 2 2 = 51 − 8 x 2 Now solve the two provided equations for x 2

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Chapter 13.9, Problem 15E

To determine

To calculate: The quantities of running shoes and basketball shoes that need to be sold to maximize the revenue where, the revenue function R(x), that gives the revenue in thousands of dollars where x1 represents the number in thousands of running shoes and x2 represents the number of basketball shoes.

R(x)=−5x12−8x22−2x1x2+42x1+102x2

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Chapter 13.9, Problem 14E

Chapter 13.9, Problem 16E

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Solution Summary: The author calculates the quantities of running shoes and basketball shoes that need to be sold to maximize the revenue.

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