1. Consider the initial value problem for the autonomous system x' = f(x), x(0) = am, where f : R\" > R" is a continuously differentiable ftmction. (a) [Bonus] Show that if ](x0) = (m) with 3 < 00, then limH |x(t)| > 00. Similarly, if a: > 00, then limt_m+ |x(t)| > oo. (Hint: suppose limH |x(t)| 74> 00, then there exists a subsequence {tk} such that x(tk) > y E R". Show that x(t) > y as t > ,6 and draw a contradiction.) (b) Suppose f satises |f(x)| S '1le +17, for some positive constants a, b and all x E R". Use part (a) and Gronwall's inequality to show that the solution x(t) to the NP x' = f (x) with x(0) = x0 satisfies |x(t)| : lxole\""' + Seal\" 1) for t E R, as long as the solution exists. (Hint: write out the integral equation satisfied by x(t) and use Gronwall's inequality on m(t) = |x(t)| + g or 11103) = |x(t)| + b E! where t Z 0.) Hence show that I(xo) = R for all x0 6 R", i.e., the solution exists globally for all x0 6 R"