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(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts $ (a) - (c) $ to sketch the graph. Check your work with a graphing device if you have one.

$ S(x) = x - \sin x $, $ 0 \leqslant x \leqslant 4\pi $

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Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 3

How Derivatives Affect the Shape of a Graph

Derivatives

Differentiation

Volume

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

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Part, i find the intervals of increase or decrease part. So f s, prime, is equal to 1 minus cosine x, which is always greater than 0. So behalf has interval from 0 to 4 pi, so f s prim is greater than 0. So, as is increasing part b so from the result in part, a the function as x has no local maximum and minimum values. Part c find the intervals of concavity and the inflection points light as from so this is equal to sine x. Let the second derivative be equal to 0, with half x is equal to 0 pi 2 pi 3 pi 4 pi and the interval from 0 to pi. Second derivative is positive, so the function is so f s x, concaves interval from pit to pi. So as propriis negative, so as is a concave tom and 2 pi to 3 pi, half second derivative is positive, so its function is concave r, 3 pi to 4 pi as proprium is negative, was function is conciptinflection. Points are pi 2 pi, 3 pi part d. Use the information from parts a to c to sketch the graph, so we can sketch the graph as follows: 0 pi, 2 pi, 3 pi, 4 pilik and by using the graphing device we can graph is a function as follows. So this is the graph of the function x, minus sine x,

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