Introduction to Mathematical Analysis I - Second Edition

116 4.3 SOME APPLICATIONS OF THE MEAN VALUE THEOREM The proof is now complete. Example 4.3.2 Let n ∈ N and consider the function f : ( 0 , ∞ ) → R given by f ( x ) = x n . Then f is differentiable and f 0 ( x ) = nx n − 1 6 = 0 for all x ∈ ( 0 , ∞ ) . It is also clear that f (( 0 , ∞ )) = ( 0 , ∞ ) . It follows from the Inverse Function Theorem that f − 1 : ( 0 , ∞ ) → ( 0 , ∞ ) is differentiable and given y ∈ ( 0 , ∞ ) ( f − 1 ) 0 ( y ) = 1 f 0 ( f − 1 ( y )) = 1 n ( f − 1 ( y )) n − 1 . Given y > 0, the value f − 1 ( y ) is the unique positive real number whose n -th power is y . We call f − 1 ( y ) the (positive) n -th root of y and denote it by n √ y . We also obtain the formula ( f − 1 ) 0 ( y ) = 1 n ( n √ y ) n − 1 . Exercises 4.3.1 (a) Let f : R → R be differentiable. Prove that if f 0 ( x ) is bounded, then f is Lipschitz continuous and, in particular, uniformly continuous. (b) Give an example of a function f : ( 0 , ∞ ) → R which is differentiable and uniformly continuous but such that f 0 ( x ) is not bounded. 4.3.2 I Let f : R → R . Suppose there exist ` ≥ 0 and α > 0 such that | f ( u ) − f ( v ) | ≤ ` | u − v | α for all u , v ∈ R . (4.8) (a) Prove that f is uniformly continuous on R . (b) Prove that if α > 1, then f is a constant function. (c) Find a nondifferentiable function that satisfies the condition above for α = 1. 4.3.3 B Let f and g be differentiable functions on R such that f ( x 0 ) = g ( x 0 ) and f 0 ( x ) ≤ g 0 ( x ) for all x ≥ x 0 . Prove that f ( x ) ≤ g ( x ) for all x ≥ x 0 . 4.3.4 Let f , g : R → R be differentiable functions satisfying (a) f ( 0 ) = g ( 0 ) = 1 (b) f ( x ) > 0, g ( x ) > 0 and f 0 ( x ) f ( x ) > g 0 ( x ) g ( x ) for all x . Prove that f ( 1 ) g ( 1 ) > 1 > g ( 1 ) f ( 1 ) .