Let f I R be differentiable at c I Establish the Straddle Lemma Given 0 there exists (e) 0 such that if u, v I satisfy c (e) u c v c (e), then we have f(v) f(u) (v u) f(c) e(v u) The (e) is given by Definition 6 1 1 Subtract and add the term f(c) cf(c) on the left side and use the Triangle Inequali...

Given 0 let 0 be such that if 0 w c w I then fw fc w cfc w c Now tak ...

Let f : I R be differentiable at c I. Establish the Straddle Lemma: Given

Question:

Let f : I → R be differentiable at c ∈ I. Establish the Straddle Lemma: Given ε > 0 there exists δ(e) > 0 such that if u, v ∈ I satisfy c - δ(e) < u < c < v < c + δ(e), then we have f(v) - f(u) - (v - u) fʹ(c)| < e(v - u). [The δ(e) is given by Definition 6.1.1. Subtract and add the term f(c) - cfʹ(c) on the left side and use the Triangle Inequality.]