Let f : R → R be a differentiable and periodic function with period T. (i) Show that there is c ∈ (0, T) such that f′(c) = 0. (ii) Prove that f′ is periodic with period T. (iii) Assume that f is twice differentiable. Show that there is ξ ∈ (0, T) such that f′′(ξ) = 0 (there is an inflexion point)

Let f : R → R be a differentiable and periodic function with period T. (i) Show that there is c ∈ (0, T) such that f′(c) = 0. (ii) Prove that f′ is periodic with period T. (iii) Assume that f is twice differentiable. Show that there is ξ ∈ (0, T) such that f′′(ξ) = 0 (there is an inflexion point)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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Question

Let f : R → R be a differentiable and periodic function with period T.
(i) Show that there is c ∈ (0, T) such that f′(c) = 0.
(ii) Prove that f′ is periodic with period T.
(iii) Assume that f is twice differentiable. Show that there is ξ ∈ (0, T)
such that f′′(ξ) = 0 (there is an inflexion point)

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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