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Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x) #0 for any r € (a, b). Determine whether there exists k = (a, b) such that f(k) - f(a)_ f'(k) g(b) – g(k) g'(k)* h: [a, b] → R defined by h(x) = f(x)g(x) - f(a)g(x) – g(b)f(r), r = [a,b], (Hint: consider the function and compute h'.)

Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x) #0 for any r € (a, b). Determine whether there exists k = (a, b) such that f(k) - f(a)_ f'(k) g(b) – g(k) g'(k)* h: [a, b] → R defined by h(x) = f(x)g(x) - f(a)g(x) – g(b)f(r), r = [a,b], (Hint: consider the function and compute h'.)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 52E
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(a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x) #0 for any z € (a, b). Determine whether there exists k € (a, b) such that f(k) - f(a) f'(k) g(b) – g(k) g' (k) (Hint: consider the function h: [a, b] → R defined by = h(x) = f(x)g(x) - f(a)g(x) - g(b)f(x), x= [a,b], and compute h'.)
Transcribed Image Text:(a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x) #0 for any z € (a, b). Determine whether there exists k € (a, b) such that f(k) - f(a) f'(k) g(b) – g(k) g' (k) (Hint: consider the function h: [a, b] → R defined by = h(x) = f(x)g(x) - f(a)g(x) - g(b)f(x), x= [a,b], and compute h'.)
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