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 ² , A = 1 2 3 1 4 , y = 2 43 . 5. Let A be a 2 × 2 symmetric matrix, x ∈ R 2 . Deﬁne 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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 Spring '12
 DouglasWard
 Math

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