Explanation Given information: The function is, f ( x , y ) = x 2 + x y + y 2 Calculation: Consider the function, f ( x , y ) = x 2 + x y + y 2 ............ (1) Partial derivative of equation (1) with respect to x . f x = d d x ( x 2 + x y + y 2 ) = 2 x + y Partial derivative of equation (1) with respect to y . f y = d d x ( x 2 + x y + y 2 ) = x + 2 y Both f x and f y are equal at ( 0 , 0 ) so, the critical point is ( 0 , 0 ) . Consider the curve, x 2 + y 2 = 1 y = 1 − x 2 Substitute 1 − x 2 for y in equation (1). f ( x , y ) = x 2 + x ( 1 − x 2 ) + ( 1 − x 2 ) 2 = x 2 + x ( 1 − x 2 ) + ( 1 − x 2 ) = x ( 1 − x 2 ) + 1 ............ (2) Differentiate equation (2) with respect to x . f ′ ( x ) = x − 2 x 2 1 − x 2 + 1 − x 2 = − x 2 + 1 − x 2 1 − x 2 = − 2 x 2 + 1 1 − x 2 ............ (3) Substitute 0 for f ′ ( x ) in equation (3). 0 = − 2 x 2 + 1 1 − x 2 − 2 x 2 + 1 = 0 x = ± 1 2 So, y = ± 1 2 So, the critical points are ( 1 2 , 1 2 ) , ( 1 2 , − 1 2 ) , ( − 1 2 , 1 2 ) ( − 1 2 , − 1 2 )

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 11.7, Problem 23PS

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The absolute extrema of f on the closed bounded set.

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