(a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x)0 for any a E (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'.) = (b) If f(x) x² and g(x) x³ with x [-1, 1], find the c € (-1, 1) by using Cauchy's Mean Value Theorem. = (c) For the functions in part (b), determine whether there exists c € (-1, 1) such that f(1)-f(-1) f'(c) a(1)-a(-1) g'(c) =

(a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x)0 for any a E (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'.) = (b) If f(x) x² and g(x) x³ with x [-1, 1], find the c € (-1, 1) by using Cauchy's Mean Value Theorem. = (c) For the functions in part (b), determine whether there exists c € (-1, 1) such that f(1)-f(-1) f'(c) a(1)-a(-1) g'(c) =

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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