432hwk1 - x R n , y R m . (a) Show that x A T y = Ax y ....

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Math 432/532 Homework 15 pts. Due January 20, 2012 432 students: Solve any 5 of the 6 problems. 532 students: Solve all 6 problems. 1. Find the local and global minimizers and maximizers of the following functions, ap- plying the theorems discussed in class. Explain your reasoning carefully. (a) f ( x ) = 2 + 2 x 2 - x 4 (b) f ( x ) = xe - x 2 (c) f ( x ) = x 2 ln x 2. Prove Taylor’s theorem with remainder (see the accompanying handout) for n = 1. Hint: Apply Rolle’s Theorem on the interval between x and x * to the function g ( t ) = f ( x ) - f ( t ) - f 0 ( t )( x - t ) - ( f ( x ) - f ( x * ) - f 0 ( x * )( x - x * ))( x - t ) 2 / ( x - x * ) 2 . 3. Find all critical points of the functions (a) f ( x, y ) = 3 x 2 + 5 xy + 2 y 2 - 9 x - 7 y (b) f ( x, y ) = e x cos y (c) f ( x, y ) = xy - x 2 y - xy 2 4. Let A be an m × n matrix,
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Unformatted text preview: x R n , y R m . (a) Show that x A T y = Ax y . (b) Check this equation with x = 5 3 , A = -1 2 3 1 4 , y = -2 4-3 . 5. Let A be a 2 2 symmetric matrix, x R 2 . Dene f ( x ) = x Ax . Show that f ( x ) = 2 Ax . 6. For f : R R , assume that f , f 00 , f (3) and f (4) exist and are continuous on the interval ( x *-1 , x * + 1). Suppose that f ( x * ) = 0, f 00 ( x * ) = 0, and f (3) ( x * ) = 0, and f (4) ( x * ) < 0 Prove that x * is a strict local maximizer of f ....
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This note was uploaded on 03/18/2012 for the course MTH 432 taught by Professor Douglasward during the Spring '12 term at Miami University.

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