1. For a continuous, differentiable function f(x), and an interval [a,b], supose that f(a) and f(b) have opposite signs and that f' does not equal 0. Show that f(x) is exactly one root between a and b.
2. Show that for any numbers a and b, the inequality |sin(b) - sin(a)| is less than or equal to |a - b| is true.