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Tutorial_11 - IELM 202 Tutorial 11 definitions of concave...

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Unformatted text preview: IELM 202 Tutorial 11 1/6/11 definitions of concave function: definitions of convex function: Formal: Given any two points x 1 , x 2 , if f( α x 1 +(1- α ) x 2 ) ≥ α f( x 1 )+(1- α )f( x 2 ), for all α , 0 ≤ α≤ 1, then f( x ) is a concave function. Given any two points x 1 , x 2 , if f( α x 1 +(1- α ) x 2 ) ≤ α f( x 1 )+(1- α )f( x 2 ), for all α , 0 ≤ α≤ 1, then f( x ) is a convex function. Less formal: If we draw a line segment linking any two point on the curve, the line is always under the curve. If we draw a line segment linking any two point on the curve, the line is always above the curve. Informal : curve downward curve upward illustration of one variable case: x f(x) x x 1 2 x f(x) x x 1 2 concave function convex function Testing whether function is concave, convex in one variable case: f(x) is concave if and only if d f(x) dx 2 2 ≤ 0, for all x. f(x) is convex if and only if d f(x) dx 2 2 ≥ 0, for all x....
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This note was uploaded on 01/05/2011 for the course IELM IELM202 taught by Professor D during the Fall '10 term at HKUST.

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Tutorial_11 - IELM 202 Tutorial 11 definitions of concave...

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