**Unformatted text preview: **∂x ∂z and ∂y ∂z at (1 , 1 , 1). (b) Use the results from (a) to determine a tangent vector to the curve γ at (1 , 1 , 1). 5. (5) Consider the matrix A = 2-1-1 1 3 Is A positive-, negative- or indeﬁnite? 6. (8) Let f ( x,y ) = x 3 y + 4 y 2 and let-→ u be a unit vector in the direction of (1 , 2). (a) Calculate the Hessian H f (1 ,-1) (b) Calculate ∂f ∂-→ u (1 ,-1) and ∂ 2 f ∂-→ u 2 (1 ,-1). 7. (5) Suppose f : R 2 → R is a diﬀerentiable function that satisﬁes f (0 , 0) = 1 and f ( x,y ) ≥ 5 for all ( x,y ) ∈ R 2 with x 2 + y 2 > 3. Explain why f must attain its minimum value. Typeset by A M S-T E X 1...

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- Spring '14
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- Optimization, differentiable function, minimum value