Define f(x)= x+kx, 0<x<2

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Define

f(x)=\sin x+kx, \qquad 0<x<2\pi,

where

k

is a real constant. (i) Show that the points

x=\dfrac{\pi}{2}

and

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

are critical points of

f

if and only if

k=-\cos\dfrac{\pi}{2}

. (ii) Using the nature of the turning points, determine the range of

k

for which

f

attains a local maximum at

x=\dfrac{\pi}{2}

and a local minimum at

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

. (iii) For the values of

k

obtained in part (ii), state whether any additional critical points occur in

(0,2\pi)

and justify your answer.