Suppose that (a, b) is an open interval, that f (a, b) R is differentiable on (a, b), and that x0 (a, b) is a proper local maximum of f (see Exercise 4 1 8) a) Prove that given 0, there exist x1 x0 x2 such that f(x1) 0, f(x2) 0, and xj x0 for j 1,2 b) Make and prove an analogous statement for a pro...

a By Exercise 418 fx 0 0 Let 0 Since fx 0 is a proper local maxi ...

Suppose that (a, b) is an open interval, that f: (a, b) R is differentiable on

Question:

Suppose that (a, b) is an open interval, that f: (a, b) → R is differentiable on (a, b), and that x0 ∈ (a, b) is a proper local maximum of f (see Exercise 4.1.8).
a) Prove that given δ > 0, there exist x1 < x0 < x2 such that fʹ(x1) > 0, fʹ(x2) < 0, and |xj - x0| < δ for j = 1,2.
b) Make and prove an analogous statement for a proper local mini¬mum.