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final_exam_review

final_exam_review - Math 20 Final Exam Review Carolyn Stein...

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Math 20 Final Exam Review Carolyn Stein Exam Date: December 13, 2011 Unconstrained Optimization Finding Stationary Points To find the stationary points or critical points , set f x = 0 , f y = 0 and find all x, y that satisfy the system of equations. A stationary point can be a local maximum , local minimum , or saddle point . Note that this can be extended to more variables - simply set every partial equal to 0 and solve the resulting system. The Extreme-Value Theorem, Classifying Stationary Points The Extreme-Value Theorem states: If f is a continuous function on a closed and bounded set S 2 R n , there exists a = h a 1 , a 2 , ..., a n i 2 S and b = h b 1 , b 2 , ..., b n i 2 S such that f ( b ) f ( x ) ( a ) for any x 2 S . In other words, an absolute maximum and absolute minimum always exist for a continuous function on a closed and bounded set. You can use the Extreme-Value Theorem to find the absolute maximums and minimums of f on S : 1. Find all stationary points of f that lie inside S 2. Find the smallest and largest values of f that lie on the boundary of S 3. These are all the possible candidates for extreme values - plug these points into f to determine which is the absolute maximum and minimum 1
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Example: Find the extreme points and extreme values of f ( x, y ) = x 2 + y 2 + y - 1 that lie inside S = { ( x, y ) : x 2 + y 2 1 } . Convex Sets A set S is convex if for any a, b 2 S , the line segment connecting a, b is com- pletely contained in S . Intiutively, it must connected, have no holes, and not “bend inwards” at any point. Mathematically, a set S in R n is convex if: x, y 2 S, λ 2 [0 , 1] ) (1 - λ ) x + λ y 2 S Example: Circle the convex sets. Concave and Convex Functions A function is concave/convex if its domain is a convex set and the line seg- ment joining any two points on the function’s graph is never above/below the graph itself. Expressed mathematically, f ( x ) defined on a convex set S is concave in S if f ((1 - λ ) a + λ b ) (1 - λ ) f ( a ) + λ f ( b ) For any a, b 2 S, λ 2 [0 , 1] Global and Local Extrema Theorem: For f ( x ) with continuous partial derivatives and convex domain S and interior point x 0 : 2
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(a) If f
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