Review solutions

# Review solutions - 1 True The constraint is a circle which...

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1. True. The constraint is a circle, which is a closed, bounded set. The EVT says the functions on closed bounded sets have a max and min. 2. False. This constraint is a parabola, which is unbounded. The EVT only holds for closed, bounded sets. 3. False. The candidates can only be on the boundary, and not the interior. 4. False. Some candidates will be where the gradient is zero, but you also need to check the boundary. 5. True. A necessary condition for an interior point to be a max/min is for the gradient to be zero. 6. False. C could just be a local max or min, or a saddle. 7. True. If A is diagonalizable, A= PDP -1 and A 57 =PD 57 P -1 so A 57 is also diagonalizable. 8. True. Can show algebraically with chain rule. 9. True. By the definition, the gradient is in the direction of maximal increase of f. 10. True. If the curves intersected, then f(x,y) would not be a function because for some point f would equal two different numbers. 11. True. Showed on previous problem set. Ax=lambda *x. Multiply both sides by A. Then A^2*x = A*lambda*x = lambda*Ax= lambda^2*x.

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