## math analysis

Midterm 2 Due 11.13.13 by 3pm. Submit the exam to me in oce 1119 of WWH. Late exams will not be accepted. 1.(20 points) Let f : Rd ! R be a C1 function such that f(ax) = akf(x) for any a 2 R where k 1 is an integer. Show that rf(x) x = kf(x) 2.(10 points) Let f(x; y) = cos(ex + 3y). Compute D2f. Remark: D2f is just the derivative of the rf. 3.(20 points) Let Rd be open. Suppose that f : ! R satises Xd j=1 @2f @x2j = 0: Let : R ! R be a C1 function and assume it is convex (also known as concave up). Show that g(x) = (f(x)) satises Xd j=1 @2g @x2j 0; when x 2 . 4.

Midterm 2 Due 11.13.13 by 3pm. Submit the exam to me in oce 1119 of WWH. Late exams will not be accepted. 1.(20 points) Let f : Rd ! R be a C1 function such that f(ax) = akf(x) for any a 2 R where k 1 is an integer. Show that rf(x) x = kf(x) 2.(10 points) Let f(x; y) = cos(ex + 3y). Compute D2f. Remark: D2f is just the derivative of the rf. 3.(20 points) Let Rd be open. Suppose that f : ! R satises Xd j=1 @2f @x2j = 0: Let : R ! R be a C1 function and assume it is convex (also known as concave up). Show that g(x) = (f(x)) satises Xd j=1 @2g @x2j 0; when x 2 . 4.(20 points) Use Taylor’s theorem to prove the expansion (x + y)n = Xn k=0 nk xn??kyk 1