Let f x = x cos - 1 ⁡ - sin ⁡ x , x ∈ - π 2 , π 2 , then which of the following…

Let fx=xcos-1⁡-sin⁡x,x∈-π2,π2, then which of the following is true?
  1. f' is increasing in -π2,0 and decreasing in 0,π2
  2. f'0=-π2
  3. f is not differentiable at x=0
  4. f' is decreasing in -π2,0 and increasing in 0,π2

Solution

f'x=xπ-cos-1sinx=x(ππ2-sin-1sinx=xπ2+x
fx=xπ2+xx≥0xπ2-xx<0
f'x=π2+2xx≥0π2-2xx<0
f'x is increasing in 0, π2 and decreasing in -π2, 0.

Asked in: JEE Main 2020 (08 Jan Shift 1)

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