Introduction to Mathematical Analysis I - Second Edition

107 where 0 < c < 1. (a) Prove that the function is differentiable on R . (b) Prove that for every α > 0, the function f 0 changes its sign on ( − α , α ) . 4.1.12 Let f be differentiable at x 0 ∈ ( a , b ) and let c be a constant. Prove that (a) lim n → ∞ n f ( x 0 + 1 n ) − f ( x 0 ) = f 0 ( x 0 ) . (b) lim h → 0 f ( x 0 + ch ) − f ( x 0 ) h = c f 0 ( x 0 ) . 4.1.13 Let f be differentiable at x 0 ∈ ( a , b ) and let c be a constant. Find the limit lim h → 0 f ( x 0 + ch ) − f ( x 0 − ch ) h . 4.1.14 Prove that f : R → R , given by f ( x ) = | x | 3 , is in C 2 ( R ) but not in C 3 ( R ) (refer to Defini- tion 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. We start with some basic definitions of minima and maxima. Recall that for a ∈ R and δ > 0, the sets B ( a ; δ ) , B + ( a ; δ ) , and B − ( a ; δ ) denote the intervals ( a − δ , a + δ ) , ( a , a + δ ) and ( a − δ , a ) , respectively. Definition 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 a ∈ D if there exists δ > 0 such that f ( x ) ≥ f ( a ) for all x ∈ B ( a ; δ ) ∩ D . Similarly, we say that f has a local (or relative) maximum at a ∈ D if there exists δ > 0 such that f ( x ) ≤ f ( a ) for all x ∈ B ( a ; δ ) ∩ D . In January 1638, Pierre de Fermat described his method for finding maxima and minima in a letter written to Marin Mersenne (1588–1648) who was considered as “the center of the world of science and mathematics during the first half of the 1600s.” His method presented in the theorem below is now known as Fermat’s Rule. 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 a ∈ I and f is differentiable at a , then f 0 ( a ) = 0.