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Unformatted text preview: 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 where f ( x ) = 0 or where f ( x ) 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 , y ), where either: 1. ∂f ∂x = ∂f ∂y = 0, or 2. f ( x , y ) 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 , y ) 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 is a point where f ( x ) = 0, we can use the second-derivative test: If f 00 ( x ) > 0 then the function is concave up and x is where a local minimum is; if f 00 ( x ) < 0 then the function is concave down and x 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 , y ), satisfying criterion 1 is where a local minimum is located, then the Hessian evaluated at ( x , y ) is positive definite , i.e. all the eigenvalues are 127 128 APPENDIX A. REVIEW OF MATHEMATICAL RESULTS strictly positive. Likewise, if (strictly positive....
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- Fall '06
- Critical Point