CBSE Class 12-science Answered

F(x) = cos(2x) - sin(2x) ........................(1)

Maximum and minimum of above function can be obtained by differentiating above function and equate to zero.

dF/dx = -2 [ sin(2x) +cos(2x) ] = 0

From above expression, we get ,  sin(2x) = - cos(2x)

From the figure , we see that sine and cosine functions have values of equal magnitude but opposite in sign at 2x=(3π)/4 and 2x=(7π)/4 .

Hence maximum and minimum values are occuring at and 2x=(7π)/4 .

At 2x=(3π)/4 , F( (3π)/4 ) = cos ( (3π)/4 ) - sin ( (3π)/4 ) = -√2

At 2x=(7π)/4 , F( (7π)/4 ) = cos ( (7π)/4 ) - sin ( (7π)/4 ) =  √2

Hence range of function " F(x) = cos(2x) - sin(2x) " is ( -√2 , √2 )