Explanation Given: The given function is f x , y = x 4 + y 4 + 4 x − 32 y + 80 .... 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 4 + y 4 + 4 x − 32 y + 80 .... Given f x = 4 x 3 + 4 f y = 4 y 3 − 32 f x x = 9 x 2 f y y = 12 y 2 f x y = 0 Equate f x and f y equal to zero

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Chapter 6, Problem 26RE

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The relative maximum and minimum values of fx,y=x4+y4+4x−32y+80 , and specify where each occurs.

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The relative maximum and minimum values of f x , y = x 4 + y 4 + 4 x − 32 y + 80 , and specify where each occurs.

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The relative maximum and minimum values of fx,y=x4+y4+4x−32y+80 , and specify where each occurs.

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