Question

Find the difference between the greatest and least values of the function f(x)=sin2x-x

Answer 1

Answer:

√3  + 2π/3

Step-by-step explanation:

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