appA - Appendix A Review of Mathematical Results A.1 Local...

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Appendix A Review of Mathematical Results A.1 Local Minimum and Maximum for Multi-Variable Functions Recall that for a single-variable function, f ( x ), local extrema (local maxima or minima) can occur only at critical points . For single-variable functions, the critical points are the points x 0 where f 0 ( x 0 ) = 0 or where f ( x 0 ) is undefined. In multivariable calculus, where there is more than one independent variable, say f ( x, y ), we have The critical points of f ( x, y ) are those points, ( x 0 , y 0 ), where either: 1. ∂f ∂x = ∂f ∂y = 0, or 2. f ( x 0 , y 0 ) is undefined. A local extrema can occur only at a critical point. Note that another way of expressing condition 1, above, is to say that the gradient of f , f , must be zero. Now consider the critical points where f ( x 0 , y 0 ) are defined (i.e. we are in condition 1, not 2). Then how can we tell if it’s a local minimum, maximum, or saddle point? Recall that for a single-variable function where x 0 is a point where f 0 ( x 0 ) = 0, we can use the second-derivative test: If f 00 ( x 0 ) > 0 then the function is concave up and x 0 is where a local minimum is; if f 00 ( x 0 ) < 0 then the function is concave down and x 0 is where a local maximum is. The equivalent test for multivariable functions is to look at the Hessian of the function f ( x, y ), which is a matrix of second partials: Hf ( x, y ) = 2 f ∂x 2 2 f ∂x∂y 2 f ∂y∂x 2 f ∂y 2 . (A.1) If the critical point, ( x 0 , y 0 ), satisfying criterion 1 is where a local minimum is located, then the Hessian evaluated at ( x 0 , y 0 ) is positive definite , i.e. all the eigenvalues are 131
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132 APPENDIX A. REVIEW OF MATHEMATICAL RESULTS strictly positive. Likewise, if ( x 0 , y 0 ) is a location of a local maximum, then the Hessian evaluated at ( x 0 , y 0 ) will be negative definite , i.e. all the eigenvalues will be strictly
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