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 ∈ R and a < 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 ≤ b and f (a1) ̸ = f (b1). By Theorem 4.2.3, there exists c ∈ (a1,b1) such that f (b1) − f (a1) f ′ (c)= ̸ = 0, b1 − a1 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) = f ′ (c) > 0. x2 − x1 This implies f (x1) < f (x2). Therefore, f is strictly increasing on (a,b). The proof of (ii) is similar. □
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