Suppose that f R R If f exists and is bounded on R, and there is an 0 0 such that f '(x) 0 for all x R, prove that there exists a 0 such that if f(x0) 0 for some x0 R, then f has a root that is, that f(c) 0 for some c R ...

Let fx M and choose r 0 0 M Let 0 be so small that ...

Suppose that f : R R. If f exists and is bounded on R, and there

Question:

Suppose that f : R → R. If f" exists and is bounded on R, and there is an ε0 > 0 such that |f'(x)| > ε0 for all x ∈ R, prove that there exists a δ > 0 such that if |f(x0)| < δ > 0 for some x0 ∈ R, then f has a root; that is, that f(c) = 0 for some c ∈ R.