At what points in the interval [0, 2π], does the function sin 2 x attain its maximum value?
The function is given as f ( x ) = sin 2 x in the interval [ 0 , 2 π ] .
Differentiate the given function,
f ′ ( x ) = 2 cos 2 x
Put f ′ ( x ) = 0 ,
2 cos 2 x = 0 cos 2 x = 0
This gives the value as,
2 x = π 2 , 3 π 2 , 5 π 2 , 7 π 2 x = π 4 , 3 π 4 , 5 π 4 , 7 π 4
As the interval is given as [ 0 , 2 π ] , so the value of the function at critical points and the end points of the interval.
At x = 0 ,
f ( 0 ) = sin 2 ( 0 ) = 0
At x = π 4 ,
f ( π 4 ) = sin 2 ( π 4 ) = sin π 2 = 1
At x = 3 π 4 ,
f ( 3 π 4 ) = sin 2 ( 3 π 4 ) = sin 3 π 2 = − 1
At x = 5 π 4 ,
f ( 5 π 4 ) = sin 2 ( 5 π 4 ) = sin 5 π 2 = 1
At x = 7 π 4 ,
f ( 7 π 4 ) = sin 2 ( 7 π 4 ) = sin 7 π 2 = − 1
At x = 2 π ,
f ( 2 π ) = sin 2 ( 2 π ) = 0
Therefore, the maximum value of f ( x ) is 1 which is at x = π 4 , 5 π 4 .
The function is given as
Differentiate the given function,
Put
This gives the value as,
As the interval is given as
At
At
At
At
At
At
Therefore, the maximum value of
