Web material
where Rr-\(x,h) is the remainder. Furthermore,mr
1
Problems
12.1. Suppose / : [0, oo) H-» K is continuous, /(0) = 0, / is differentiable on
(0, oo) and /' is nondecreasing. Prove that g(x) — ^p- is nondecreasing for
x > 0.12.2. Let A C R" be an open convex set and / : A H-J- IRm be differentiable.
If f'(t) = 0, Vi then show that / is constant.12.3. Compute the second order Taylor's formula for f(x,y) = sin(x + 2y)
around the origin.12.4. Let feC^2 and x e _." be local minimizer.
a) Prove the first order necessary condition (x is a local minimizer then
V/(x) = 6) using Taylor's approximation.
b) Prove the second order necessary condition (x is a local minimizer then
V^2 /(x*) is positive semi-definite) using Taylor's approximation.
c) Design an iterative procedure to find V/(x) = 8 in such a way that it
starts from an initial point and updates as x& = Xk-\ + Pk- The problem at
each iteration is to find a direction Pk that makes V/(xfc_i) closer to the null
vector. Use the second order Taylor's approximation to find the best pk at
any iteration.
d) Use the above results to find a local solution to
min/(x 1 ,x 2 ) = x\ + 2x\ + 2Ax\ + x\ + Y2x\.Start from [1, if.
Web material
http://archives.math.utk.edu/visual.calculus/3/index.html
http://calclab.math.tamu.edU/~belmonte/ml51/L/c5/L53.pdf
http://ccrma-www.stanford.edu/"jos/mdft/Formal_Statement_Taylor_s_
Theorem.html
http://courses.math.nus.edu.sg/mall04/lecture_notes/Notes_l.pdf
http://d.faculty.umkc.edu/delawarer/RDvsiCalcList.htm
http://en.wikipedia.org/wiki/Derivative
http://en.wikipedia.org/wiki/L'Hopital's_rule