hw4 - f g + g f. Prove also that 1 f =-1 f 2 f, whenever f...

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Homework 4 – Math 118C, Spring 2010 Due on Tuesday, April 27th, 2010 1. Give a matrix A R n × m , prove that (a) k A k 1 = max 1 j m n i =1 | a ij | . (b) k A k 2 = max ± λ | λ is an eigenvalue of A T A ² . (c) k A k = max 1 i n m j =1 | a ij | . (d) k A k 2 ≤ k A k F n k A k 2 , where k A k 2 F = i,j a 2 ij . (e) max i,j | a i j | ≤ k A k 2 mn max i,j | a ij | . (f) 1 m k A k ≤ k A k 2 n k A k . (g) 1 n k A k 1 ≤ k A k 2 m k A k 1 . 2. If f (0 , 0) = 0 and f ( x, y ) = xy x 2 y 2 , ( x, y ) 6 = (0 , 0) , prove that D 1 f ( x, y ) and F 2 f ( x, y ) exist at every point in R 2 , although f is not continuous at (0 , 0). 3. Suppose that f is a real-valued function defined on an open set E R n , and that the partial derivatives D 1 f, . . . , D n f are bounded in E . Prove that f is continuous in E . 4. Suppose that f is a differentiable real function in an open set E R n , and that f has a local maximum at a point x E . Prove that df ( x ) = 0. 5. If f and g are differentiable real functions in R n , prove that ( fb ) =
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Unformatted text preview: f g + g f. Prove also that 1 f =-1 f 2 f, whenever f 6 = 0. 1 2 6. Suppose f is a dierentiable mapping of R into R 3 such that k f ( t ) k 2 = 1 for every t . Prove that f ( t ) f ( t ) = 0. 7. Dene f (0 , 0) = 0, and f ( x, y ) = x 2 + y 2-2 x 2 y-4 x 6 y 2 ( x 4 + y 2 ) 2 , ( x, y ) 6 = (0 , 0) . (a) Prove, for all ( x, y ) R 2 , that 4 x 4 y 2 ( x 4 + y 2 ) 2 . Conclude that f is continuous. (b) For 0 2 ,- < t < , dene g ( t ) = f ( t cos , t sin ) . Show that each g has a strict local minimum at t = 0. (c) Show that (0 , 0) is not a local minimum for f ....
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hw4 - f g + g f. Prove also that 1 f =-1 f 2 f, whenever f...

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