Introduction to Mathematical Analysis I - 3rd Edition

102 4.3 Some Applications of the Mean Value Theorem 4.3 Some Applications of the Mean Value Theorem In this section, we assume that a,b∈Randa<b. In the proposition below, we show that it is possible to use the derivative to determine whether a function is constant. The proof is based on the Mean Value Theorem. Proposition 4.3.1 Let f be continuous on [a,b] and differentiable on (a,b). If f ′(x) =0 for all x ∈(a,b), then f is constant on[a,b]. Proof: Suppose by contradiction that f is not constant on [a,b]. Then there exist a1 and b1 such that a≤a1 <b1 ≤band f (a1)̸ =f (b1). By Theorem 4.2.3, there exists c ∈(a1,b1) such that f ′(c) = f (b1)−f (a1) b1 −a1̸ =0, which is a contradiction. Therefore, f is constant on[a,b]. □ The next application of the Mean Value Theorem concerns developing simple criteria for monotonicity of real-valued functions based on the derivative. Figure 4.5: Strictly Increasing Function. Proposition 4.3.2 Let f be differentiable on(a,b). (i) If f ′(x) >0 for all x ∈(a,b), then f is strictly increasing on (a,b). (ii) If f ′(x) <0 for all x ∈(a,b), then f is strictly decreasing on (a,b). Proof: Let us prove (i). Fix any x1,x2 ∈(a,b) with x1 <x2. By Theorem 4.2.3, there exists c ∈(x1,x2) such that f (x2)−f (x1) x2 −x1 =f ′(c) >0. This implies f (x1)<f (x2). Therefore, f is strictly increasing on(a,b). The proof of (ii) is similar. □

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