Advanced Mathematics

1. Let a, b, c be integers. Suppose that a divides bc and that (a, b) = 1. Prove that a divides c

Problem 5 Classify (local and global) the stationary points of f: R² → R, ƒ(x) = x¹Qx+b¹x where a/2 Q = (a/2 °1²) = and b = (1, 2) for a € R. Consider all possibilities depending on a.

Problem 6 Let A € Rmxn, m ≤ n, rank(A) = m, and c = R¹, b = Rm. Show that the problem minimize ||xc|| s.t. Ax = b has a unique solution and find the expression for x*.

Any countably infinite product of seprable spaces is separable. This Question shows that the box topology does not behave

Define an equivalence relation on S2 = {(x, y, z) | x2+y2++z2 = 1} by defining the equivalence classes to be {

Consider the One-dimensional Wave Equation for vibrations on a string of length L with a free end and linear damping.

Q 11 Use the graph to find: (a) The numbers, if any, at which f has a local maximum. What are these local maxima? (b) The numbers, if any, at which f has a local minimum. What are these local minima?

8. Suppose that the manufacturer of a gas clothes dryer has found that when the unit price is p dollars, the revenue R (in dollars) is R(p) = -4p² +8,000p. (a) At what prices p is revenue zero? (b) For what range of prices will revenue exceed $1,000,000?

Assume a continuous function f(x) defined on x axis with a uniform grid of spacing h. Using appropriate Taylor series expansions2, find the leading order

2. Where on the path r(t) = (t²-5t)i + (21+1)] +38² k vectors orthogonal? are the velocity and acceleration