# hw4 - f ∇ g g ∇ f Prove also that ∇ ³ 1 f ´ =-1 f 2...

This preview shows pages 1–2. Sign up to view the full content.

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 ) =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f ∇ g + g ∇ f. Prove also that ∇ ³ 1 f ´ =-1 f 2 ∇ f, whenever f 6 = 0. 1 2 6. Suppose f is a diﬀerentiable 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. Deﬁne 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 ≤ θ ≤ 2 π ,-∞ < t < ∞ , deﬁne g θ ( t ) = f ( t cos θ, t sin θ ) . Show that each g θ has a strict local minimum at t = 0. (c) Show that (0 , 0) is not a local minimum for f ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

hw4 - f ∇ g g ∇ f Prove also that ∇ ³ 1 f ´ =-1 f 2...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online