The difference between the greatest and least value of the function f (x) = sin 2x - x ona)andpi;/2b)andpi;c)3andpi;/2d)andpi;/4Correct answer is option 'B'. Can you explain this answer?

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The difference between the greatest and least value of the function f (x) = sin 2x - x on

a)
π/2
b)
π
c)
3π/2
d)
π/4

Correct answer is option 'B'. Can you explain this answer?

FREE This question is part of

JEE Advanced Level Test: Relations and Functions- 4

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The difference between the greatest and least value of the function f ...

Method to Solve :

f(x)=sin2x-x

f'(x) = 2Cos2x - 1

putting f'(x) = 0

=> 2Cos2x - 1 = 0

=> 2Cos2x = 1

=> Cos2x = 1/2

=> 2x = π/3   , 5π/3

=> x = π/6   , 5π/6

f''(x) = 2(-2)Sin2x

f''(x) = - 4Sin2x

for x = π/6   f''(x) is - ve

=> x = π/6 will give max value

for x = 5π/6   f''(x) is + ve

=> x = 5π/6 will give min value

Greatest Value = Sin( π/3) - π/6

Least Value = Sin(5π/3) - 5π/6

difference between the greatest and least values of the function

= Sin( π/3) - π/6 - Sin(5π/3) + 5π/6

= √3 + 4π/6

= √3 + 2π/3

if Range of x is -π/2 to π/2

then 2x = π/3   & -π/3

F(x) Max = Sin(π/3) - π/6

F(x) Min = Sin(-π/3) + π/6

Difference = 2Sin(π/3) - 2π/6

= √3 - π/3

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The difference between the greatest and least value of the function f (x) = sin 2x - x ona)π/2b)πc)3π/2d)π/4Correct answer is option 'B'. Can you explain this answer?

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