Introduction to Mathematical Analysis I - Second Edition

114 4.3 SOME APPLICATIONS OF THE MEAN VALUE THEOREM Prove that the equation n ∑ k = 0 a k cos ( 2 k + 1 ) x = 0 has a solution on ( 0 , π 2 ) . 4.2.7 Let f : [ 0 , ∞ ) → R be a differentiable function. Prove that if both lim x → ∞ f ( x ) and lim x → ∞ f 0 ( x ) exist, then lim x → ∞ f 0 ( x ) = 0 4.2.8 B Let f : [ 0 , ∞ ) → R be a differentiable function. (a) Show that if lim x → ∞ f 0 ( x ) = a , then lim x → ∞ f ( x ) x = a . (b) Show that if lim x → ∞ f 0 ( x ) = ∞ , then lim x → ∞ f ( x ) x = ∞ . (c) Are the converses in part (a) and part (b) true? 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 0 ( 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 a 1 and b 1 such that a ≤ a 1 < b 1 ≤ b and f ( a 1 ) 6 = f ( b 1 ) . By Theorem 4.2.3 , there exists c ∈ ( a 1 , b 1 ) such that f 0 ( c ) = f ( b 1 ) − f ( a 1 ) b 1 − a 1 6 = 0 , which is a contradiction. The next application of the Mean Value Theorem concerns developing simple criteria for monotonicity of real-valued functions based on the derivative. Proposition 4.3.2 Let f be differentiable on ( a , b ) . (i) If f 0 ( x ) > 0 for all x ∈ ( a , b ) , then f is strictly increasing on ( a , b ) . (ii) If f 0 ( x ) < 0 for all x ∈ ( a , b ) , then f is strictly decreasing on ( a , b ) . Proof: Let us prove (i). Fix any x 1 , x 2 ∈ ( a , b ) with x 1 < x 2 . By Theorem 4.2.3 , there exists c ∈ ( x 1 , x 2 ) such that f ( x 2 ) − f ( x 1 ) x 2 − x 1 = f 0 ( c ) > 0 . This implies f ( x 1 ) < f ( x 2 ) . Therefore, f is strictly increasing on ( a , b ) . The proof of (ii) is similar.