homework6sol - 32A HOMEWORK 6 SOLUTIONS MAXIMA MINIMA AND...

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32A - HOMEWORK 6 SOLUTIONS MAXIMA, MINIMA AND SADDLE POINTS PROBLEM 2.8. The distance from ( x, y )to one of the points ( a k ,b k )is d k = p ( x a k ) 2 +( y b k ) 2 , thus the function to be minimized is f ( x, y )= d 2 1 + d 2 n + ... + d 2 n =( x a 1 ) 2 x a 2 ) 2 + x a n ) 2 y b 1 ) 2 y b 2 ) 2 + y b n ) 2 . (1) The equations for critical points are ∂f ( x, y ) ∂x =2( x a 1 )+2( x a 2 )+ +2( x a n )=0 ( x, y ) ∂y y b 1 y b 2 y b n . (2) Solution: x = a 1 + a 2 + + a n n ,y = b 1 + b 2 + + b n n . PROBLEM 2.9. Taking derivatives in (2) we obtain 2 f ( x, y ) 2 =2 n, 2 f ( x, y ) ∂x∂y =0 , 2 f ( x, y ) 2 n (3) so that the Hessian is H ( x, y )=4 n 2 . This and the Frst equality (3) imply that every critical point must be a minimum. PROBLEM 2.10. The function to be minmimized in ( a, b f ( a, b )=( ax 1 + b y 1 ) 2 ax 2 + b y 2 ) 2 + ax n + b y n ) 2 . We have ( a, b ) ∂b ( ax 1 + b y 1 )+ 2( ax 2 + b y 2 +2 ( ax n + b y n ) ( a, b ) ∂a x 1 ( ax 1 + b y 1 )+2 x 2 ( ax 2 + b y 2 x n ( ax n + b y n ) thus ( a, b )must satisfy the system ( ax 1 + b y 1 )+ ( ax 2 + b y 2 ax n + b y n x 1 ( ax 1 + b y 1 x 2 ( ax 2 + b y 2 + x n ( ax n + b y n . (1) PROBLEM 2.11. This system can be rearranged in the form ( x 1 + x 2 + + x n ) a + nb = y 1 + y 2 + + y n ( x 2 1 + x 2 2 + + x 2 n ) a x 1 + x 2 + + x n ) b = x 1 y 1 + x 2 y 2 + + x n y n . (2) 1
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HOMEWORK 6 SOLUTIONS Ë MAXIMA, MINIMA AND SADDLE POINTS ® PROBLEMS 2.1, 2.2 f @ x _ , y _ D = 3 * x^2 + 4 * x * y + 5 * y^2 3 x 2 + 4 x y + 5 y 2 f1 @ x _ , y _ D = D @ f @ x, y D , x D 6 x + 4 y f2 @ x _ , y _ D = D @ f @ x, y D , y D 4 x + 10 y Solve @8 f1 @ x, y D == 0, f2 @ x, y D == 0 < , 8 x, y <D 88 x Ø 0, y Ø 0 << H @ x _ , y _ D = D @ f @ x, y D , x, x D * D @ f @ x, y D , y, y D - D @ f @ x, y D , x, y D ^2 44 f11 @ x _ , y _ D = D @ f @ x, y D , x, x D 6 Local minimum. Graph:
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Plot3D @ f @ x, y D , 8 x, - 1, 1 < , 8 y, - 1, 1 <D f @ x _ , y _ D = 3 * x^2 - 5 * y^2 + 12 * x - 10 * y 12 x + 3 x 2 - 10 y - 5 y 2 f1 @ x _ , y _ D = D @ f @ x, y D , x D 12 + 6 x f2 @ x _ , y _ D = D @ f @ x, y D , y D - 10 - 10 y Solve @8 f1 @ x, y D == 0, f2 @ x, y D == 0 < , 8 x, y <D 88 x Ø - 2, y Ø - 1 << H @ x _ , y _ D = D @ f @ x, y D , x, x D * D @ f @ x, y D , y, y D - D @ f @ x, y D , x, y D ^2 - 60 Saddle point. Graph: 2 HW6sol.nb
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Plot3D @ f @ x, y D , 8 x, - 3, - 1 < , 8 y, - 2, 0 <D f @ x _ , y _ D = x^2 * y^2 - y - y + x 2 y 2 f1 @ x _ , y _ D = D @ f @ x, y D
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homework6sol - 32A HOMEWORK 6 SOLUTIONS MAXIMA MINIMA AND...

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