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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern