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). (4.4) 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 (4.4). □ 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 a if 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 . (4.5) (ii) We say that f has a left derivative at b if 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 functiong: [a,b] →Rby g(x)=f (x)−λx.
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