Explanation Given: The parabola is y = x 2 and indicated point is ( − 3 , 0 ) . Formula used: The distance formula is given as: d = ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 Calculation: Consider ( x , y ) to be an arbitrary point on the parabola y = x 2 . The distance from ( x , y ) to ( − 3 , 0 ) using distance formula is given as, d = ( x + 3 ) 2 + ( y − 0 ) 2 First minimize the function of square of the distance given by, f ( x , y ) = ( x + 3 ) 2 + ( y − 0 ) 2 Subject to constraint y − x 2 = 0 . Now, find ∇ f ( x , y ) and ∇ g ( x , y ) . ∇ f ( x , y ) = f x ( x , y ) i + f y ( x , y ) j = ( 2 x + 6 ) i + 2 y j And, ∇ g ( x , y ) = g x ( x , y ) i + g y ( x , y ) j = − 2 x i + j According to Lagrange multiplier, ∇ f ( x , y ) = λ ∇

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Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 13.10, Problem 24E

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To calculate: The minimum distance from the parabola, y=x2 to the point (−3,0) by using Lagrange multiplier.

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Chapter 13.10, Problem 23E

Chapter 13.10, Problem 25E

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The minimum distance from the parabola, y = x 2 to the point ( − 3 , 0 ) by using Lagrange multiplier.

Solution Summary: The author calculates the minimum distance from sqrt5 to the indicated point by using Lagrange multiplier.

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To calculate: The minimum distance from the parabola, y=x2 to the point (−3,0) by using Lagrange multiplier.

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