99 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∈Rwitha<b. Suppose f andgare continuous on [a,b] and differentiable on (a,b). Then there exists c ∈(a,b) such that [ f (b)−f (a)]g′(c) = [g(b)−g(a)] f ′(c). (4.5) Proof: Define h(x) = [ f (b)−f (a)]g(x)−[g(b)−g(a)] f (x) for x ∈[a,b]. Thenh(a) =f (b)g(a)−f (a)g(b) =h(b), andhsatisfies the assumptions of Theorem 4.2.2. Thus, there exists c ∈(a,b) such that h′(c) =0. Since h′(x) = [ f (b)−f (a)]g′(x)−[g(b)−g(a)] f ′(x), this implies (4.5). □ 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 let f : [a,b] →R. (i) We say that f has a right derivative at aif lim x→a+ f (x)−f (a) x−a exists. In this case we write f ′ +(a) = lim x→a+ f (x)−f (a) x−a . (4.6) (ii) We say that f has a left derivative at bif lim x→b− f (x)−f (b) x−b exists. In this case we write f ′ −(b) = lim x→b− f (x)−f (b) x−b . (iii) We say that f is differentiable on[a,b] if f ′(x) exists for each x ∈(a,b) and, in addition, both f ′ +(a) and f −′ (b) exist. Theorem 4.2.5 — Intermediate Value Theorem for Derivatives. Let a,b∈Rwitha<b. Suppose f is differentiable on [a,b] and f ′ +(a) <λ <f ′ −(b). Then there exists c ∈(a,b) such that f ′(c) =λ. Proof: Define the function g: [a,b] →Rby g(x) =f (x)−λx.
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