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If f(x)= $\cos 2 x cos 2 x sin 2 x - cosx cosx - sinx sinx sinx cosx,$ f(x) attains its maxima at x = [[1]].

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Detailed Solution

By expanding the function, we get
f(x) = f(x)=cos2x(cosx × cosx +sinx ×sinx)−cos2x(−cosx × cosx + sinx × sinx)+sin2x(−cosx × sinx – cosx × sinx)
⇨f(x) = cos2x (cos2x + sin2x) – cos 2x(-cos2x + sin2x) + sin2x(-2cosxsinx)
Since, cos2x + sin2x = 1, cos2x-sin2x = cos2x ;
Equation becomes : f(x) = cos2x + cos2x – sin22x
⇨f(x) = cos2x + cos4x
For getting minima and maxima, put f’(x) = 0
⇨f’(x)= -2sin2x -4(2sin2xcos2x) = -2sin2x(1+4cos2x) = 0 (1)
Also, for maxima f’’(x)<0
f’’(x) = -4cos2x – 16cos4x
the equation (1) gives us 2 values x = 0, and cos2x = -1/4
putting x = 0, f’’(x) = -4cos2(0) -16cos4(0) = -20
thus x = 0, gives the maxima of the function.

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