Math 230 Final ExamFall Quarter 2013Page 3 of 17PART BSuppose thatf:R2→Rhas continuous second partial derivatives.(a) The mixed partial derivativesfxy(a, b)andfyx(a, b)are equal.
(b) There exists such a functionfwhose linear approximation at(0,0)isL(x, y) = 4x+ 2y-3and whose quadratic approximation at(0,0)isQ(x, y) = 4x2+ 2y2-3.
(c) Iffx(0,0) =fy(0,0) = 0andfxx(0,0)is positive, thenfhas a localminimum at(0,0).
(d) Iffx(0,0) =fy(0,0) = 0andfxx(0,0)·fyy(0,0)<[fxy(0,0)]2thenfhas a local minimum at(0,0).
For the next two statements, consider the following table.(x, y)ffxfyfxxfxyfyy(2,3)unknown001-36(-7,8)-600-4unknown7(e) A local minimum offoccurs at(2,3).