95 where 0 < c < 1. (a) Prove that the function is differentiable on R. ′ (b) Prove that for every α > 0, the function f changes its sign on (−α,α). 4.1.12 Let f be differentiable at x0 ∈ (a,b) and let c be a constant. Prove that 1 (a) limn→∞ n f (x0 + ) − f (x0) = f ′ (x0). n f (x0 + ch) − f (x0) (b) limh →0 = cf ′ (x0). h 4.1.13 Let f be differentiable at x0 ∈ (a,b) and let c be a constant. Find the limit f (x0 + ch) − f (x0 − ch) lim . h→0 h 4.1.14 Prove that f : R → R, given by f (x)= |x|3, is in C2(R) but not in C3(R) (refer to Defnition 4.1.3). (Hint: the key issue is differentiability at 0.) 4.2 The Mean Value Theorem In this section, we focus on the Mean Value Theorem, one of the most important tools of calculus and one of the most beautiful results of mathematical analysis. The Mean Value Theorem we study in this section was stated by the French mathematician Augustin Louis Cauchy (1789–1857), which follows from a simpler version called Rolle’s Theorem. An important application of differentiation is solving optimization problems. A simple method for identifying local extrema of a function was found by the French mathematician Pierre de Fermat (1601–1665). Fermat’s method can also be used to prove Rolle’s Theorem. Defnition 4.2.1 Let D be a nonempty subset of R and let f : D → R. We say that f has a local (or relative) minimum at c ∈ D if there exists δ > 0 such that f (x) ≥ f (c) for all x ∈ (c− δ ,c + δ ) ∩ D. Similarly, we say that f has a local (or relative) maximum at c ∈ D if there exists δ > 0 such that f (x) ≤ f (c) for all x ∈ (c− δ ,c + δ ) ∩ D. Theorem 4.2.1 — Fermat’s Rule. Let I be an open interval and f : I → R. If f has a local minimum or maximum at c ∈ I and f is differentiable at c, then f ′ (c)= 0. Proof: Suppose f has a local minimum at c. Then there exists δ > 0 suffciently small such that f (x) ≥ f (c) for all x ∈ (c − δ ,c+ δ ) ⊂ I. Since (c,c+ δ ) is a subset of (c − δ ,c + δ ), we have f (x) − f (c) ≥ 0 for all x ∈ (c,c+ δ ). x − c
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