If f x = cos ⁡ 2 x cos ⁡ 2 x sin ⁡ 2 x - cos ⁡ x cos ⁡ x - sin ⁡ x sin…

If fx=cos⁡2xcos⁡2xsin⁡2x-cos⁡xcos⁡x-sin⁡xsin⁡xsin⁡xcos⁡x, then
  1. f′x=0 at exactly three points in (-π,π)
  2. f(x) attains its maxima at x=0
  3. f(x) attains its minima at x=0
  4. f′x=0 at more than three points in (-π,π)

Solution

Expansion of determinant

fx=cos ⁡2x+cos ⁡4x

f'(x)= −2sin 2x−4sin 4x=−2sin 2x(1+4cos 2x)

∴    maxima at x=0

f'x=0  ⇒    sin 2 x = 0 ∴ x = 0 , π 2 , - π 2
and cos⁡2x= -14 has 4 solutions in - π , π

∴ f  ′ x = 0 has more than three solutions

Asked in: JEE Advanced 2017 (Paper 2)

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