(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) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(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) 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)
g(1) - g(-1)
=
f'(c)
g'(c)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F60abd4de-ba2d-4a53-b8fc-e1d2e0799cec%2F5f40695b-8084-484b-a8f8-0b7580163738%2Faecy30s_processed.jpeg&w=3840&q=75)
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 x = (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)
g(1) - g(-1)
=
f'(c)
g'(c)
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