Explanation Given: The given function is f x , y = x 3 − 6 x y + y 2 + 6 x + 3 y − 1 5 .... 1 Concept used: With the help of D-Test the relative maximum and minimum values of function of two variable can be evaluated with the help of following steps: Evaluate f x , f y , f x x , f y y and f x y and then equate f x and f y equal to 0 to find the critical point P a , b of the function. Evaluate D where D = f x x a , b ⋅ f y y a , b − f x y a , b 2 . At last the theorem states: (a) f has a maximum at point a , b if D > 0 and f x x a , b < 0 . (b) f has a minimum at a , b if D > 0 and f x x a , b > 0 . (c) f has neither a maximum nor a minimum at a , b if D < 0 . The function has a saddle point at a , b . (d) This test is not applicable if D = 0 . Calculation: With the help of concept stated above the maximum and minimum value of the function is evaluated as: Evaluate f x , f y , f x x , f y y and f x y with the help of (1) and then equate f x and f y equal to 0 to find the critical point of the function. f x , y = x 3 − 6 x y + y 2 + 6 x + 3 y − 1 5 .... Given f x = 3 x 2 − 6 y + 6 f y = − 6 x + 2 y + 3 f x x = 6 x f y y = 2 f x y = − 6 Equate f x and f y equal to zero. f x = 0 f y = 0 3 x 2 − 6 y + 6 = 0 − 6 x + 2 y + 3 = 0 x 2 − 2 y + 2 = 0 y = 3 x − 1

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Chapter 6, Problem 24RE

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The relative maximum and minimum values of fx,y=x3−6xy+y2+6x+3y−15 , and specify where each occurs.

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Chapter 6, Problem 23RE

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The relative maximum and minimum values of f x , y = x 3 − 6 x y + y 2 + 6 x + 3 y − 1 5 , and specify where each occurs.

Solution Summary: The author explains how the relative maximum and minimum values of two variables can be evaluated with the help of D-Test.

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The relative maximum and minimum values of fx,y=x3−6xy+y2+6x+3y−15 , and specify where each occurs.

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