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 ∈ 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 ′ (c)=[g(b) − g(a)] f ′ (c). (4.4) Proof: Defne 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 satisfes 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] satisfes 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 defnitions. Defnition 4.2.2 Let a,b ∈ R, a < b, and let f : [a,b] → R. f (x) − f (a) (i) We say that f has a right derivative at a if lim exists. In this case we write x→a+ x − a f (x) − f (a) ′ f (a)= lim . (4.5) + x→a+ x − a f (x) − f (b) (ii) We say that f has a left derivative at b if lim exists. In this case we write x→b− x − b f (x) − f (b) ′ f − (b)= lim . x→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 ∈ R with a < 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: Defne the function g: [a,b] → R by g(x)= f (x) − λ x.
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