Question 3 (a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x) #0 for any re(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(z) = f(r)g(r) – f(a)g(2) – g(b)f(z), 2 € a,b], and compute h'.) (b) If f(x) = z² and g(x) 23 with a [-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) g(1) g(-1) g'(c) =
Question 3 (a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b), and g'(x) #0 for any re(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(z) = f(r)g(r) – f(a)g(2) – g(b)f(z), 2 € a,b], and compute h'.) (b) If f(x) = z² and g(x) 23 with a [-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) g(1) g(-1) g'(c) =
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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![Question 3
(a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b),
and g'(x) #0 for any x € (a, b). Determine whether there exists k € (a, b)
that
such
ƒ(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(x), x= [a,b],
(Hint: consider the function
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)
g(1)-g(-1) g'(c)
=
●
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Transcribed Image Text:Question 3
(a) Suppose the functions f and g are continuous on [a, b], differentiable on (a, b),
and g'(x) #0 for any x € (a, b). Determine whether there exists k € (a, b)
that
such
ƒ(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(x), x= [a,b],
(Hint: consider the function
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)
g(1)-g(-1) g'(c)
=
●
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