Amongst all these serious posts concerning the stress of university and other such things, I believe it’s high time that I lighten things up — and what better way to lighten things up than with some math?
As promised, here’s the solution I gave to Problem #5 on the MATH 245 midterm. The problem was:
Let be points in . Show that there is a unique point which minimizes the sum , and find a formula for this point in terms of .
I was pressed for time on the exam, so since I couldn’t see any obvious solution of the problem by linear algebraic methods, I resorted (as I often do with problems like this) to tedious analytic methods. The general idea of it is as follows. You suppose , and define a function of two variables . You then take the partial derivatives and which turn out to be and respectively. Setting these both equal to zero we obtain that which happens to be the centroid of the (possibly degenerate, and certainly degenerate if ) simplex formed by those points in the plane.
When I reached this point in the solution, the exam was over, so I lost a few marks. To complete this solution, you have to take second derivatives and thereafter form the Hessian matrix of the function . You then must show that the Hessian matrix is positive-definite in order to successfully prove that is indeed a local minimum of , and is hence the minimizing point we seek.