Introduction to Mathematical Analysis I - Second Edition

111 A more general result which follows directly from the Mean Value Theorem is known as Cauchy’s Theorem. Theorem 4.2.4 — Cauchy’s Theorem. Let a , b ∈ R with a < b . Suppose f and g are continuous on [ a , b ] and differentiable on ( a , b ) . Then there exists c ∈ ( a , b ) such that [ f ( b ) − f ( a )] g 0 ( c ) = [ g ( b ) − g ( a )] f 0 ( c ) . (4.6) Proof: Define h ( x ) = [ f ( b ) − f ( a )] g ( x ) − [ g ( b ) − g ( a )] f ( x ) for x ∈ [ a , b ] . Then h ( a ) = f ( b ) g ( a ) − f ( a ) g ( b ) = h ( b ) , and h satisfies the assumptions of Theorem 4.2.2 . Thus, there exists c ∈ ( a , b ) such that h 0 ( c ) = 0. Since h 0 ( x ) = [ f ( b ) − f ( a )] g 0 ( x ) − [ g ( b ) − g ( a )] f 0 ( x ) , this implies ( 4.6 ) . The following theorem shows that the derivative of a differentiable function on [ a , b ] satisfies the intermediate value property although the derivative function is not assumed to be continuous. To give the theorem in its greatest generality, we introduce a couple of definitions. Definition 4.2.2 Let a , b ∈ R , a < b , and f : [ a , b ] → R . If the limit lim x → a + f ( x ) − f ( a ) x − a exists, we say that f has a right derivative at a and write f 0 + ( a ) = lim x → a + f ( x ) − f ( a ) x − a . If the limit lim x → b − f ( x ) − f ( b ) x − b exists, we say that f has a left derivative at b and write f 0 − ( b ) = lim x → b − f ( x ) − f ( b ) x − b . We will say that f is differentiable on [ a , b ] if f 0 ( x ) exists for each x ∈ ( a , b ) and, in addition, both f 0 + ( a ) and f 0− ( b ) exist. Theorem 4.2.5 — Intermediate Value Theorem for Derivatives. Let a , b ∈ R with a < b . Sup- pose f is differentiable on [ a , b ] and f 0 + ( a ) < λ < f 0 − ( b ) . Then there exists c ∈ ( a , b ) such that f 0 ( c ) = λ .