Explanation To verify the provided statement, consider a function, f ( x , y ) = ( x 2 − 1 ) 2 + ( y − x − 1 ) 2 This function is continuous for all x and y as ( x 2 − 1 ) and ( y − x − 1 ) is continuous for all x and y , sum or difference of squares of continuous function is a continuous function Now, apply the second partials test on this function to find the relative extrema. The partial test is: Let f have continuous second partial derivatives on an open region containing a point ( a , b ) for which f x ( a , b ) = 0 and f y ( a , b ) = 0 . Then the second derivative test discriminant is defined as: d = f x x ( a , b ) f y y ( a , b ) − [ f x y ( a , b ) ] 2 If d > 0 and f x x ( a , b ) > 0 , then f has a relative minimum at ( a , b ). If d > 0 and f x x ( a , b ) < 0 , then f has a relative maximum at ( a , b ). If d < 0, then ( a , b , f ( a , b )) is a saddle point. The test is inconclusive if d = 0. Thus, first order partial derivatives of this function will be f x = ∂ ∂ x [ ( x 2 − 1 ) 2 + ( y − x − 1 ) 2 ] = 2 ( x 2 − 1 ) × 2 x + 2 ( y − x − 1 ) × ( − 1 ) = 4 x ( x 2 − 1 ) − 2 ( y − x − 1 ) And, the derivative with respect to y : f y = ∂ ∂ y [ ( x 2 − 1 ) 2 + ( y − x − 1 ) 2 ] = 0 + 2 ( y − x − 1 ) = 2 ( y − x − 1 ) Now, to find the critical points, equate the first order partial derivatives to zero. Then, 4 x ( x 2 − 1 ) − 2 ( y − x − 1 ) = 0 …...…... (1) And, 2 ( y − x − 1 ) = 0 …...…..

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Chapter 13.8, Problem 58E

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Whether the statement “If f is continuous for all x and y, has two relative minima, then f must have at least one relative maximum” is true or false.

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Chapter 13.8, Problem 57E

Chapter 13.9, Problem 1E

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Whether the statement “If f is continuous for all x and y , has two relative minima , then f must have at least one relative maximum ” is true or false.

Solution Summary: The author explains whether the statement "If f is continuous for all , x, and y, has two relative minima, then must have at least one relative maximum" is false.

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Whether the statement “If f is continuous for all x and y, has two relative minima, then f must have at least one relative maximum” is true or false.

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Chapter 13 Solutions