Mathematics

Problem

(Conceptual reasoning on extrema positions) a. (i) Define what is meant by a zero, a maximum point and a minimum point of a real-valued function. b. (ii) Explain why the graph of

y=\sin x

attains its maximum value at

x=\dfrac{\pi}{2}+2\pi n

and its minimum value at

x=\dfrac{3\pi}{2}+2\pi n

, where

n\in\mathbb Z

. c. (iii) Compare the positions of maxima and minima of

y=\cos x

with those of

y=\sin x

and give a brief reason for any horizontal displacement that you observe. d. (iv) Sketch one complete period of

y=\sin x

and

y=\cos x

on the same set of axes, clearly marking their zeros, maxima and minima.

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