If f(x)=sin−1(cosx)cos−1(sinx)∀x∈[0,2π], then which of the following statement(s) is/are correct?
A
f(x) is differentiable in x∈[0,2π]
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B
Range of f(x) is [−π24,π24]
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C
x=π is a point of global minima as well as local minima.
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D
π∫0f(x)dx=0
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Solution
The correct options are B Range of f(x) is [−π24,π24] Cx=π is a point of global minima as well as local minima. Dπ∫0f(x)dx=0 Given : f(x)=sin−1(cosx)cos−1(sinx) ⇒f(x)=(π2−cos−1(cosx))(π2−sin−1(sinx))⇒f(x)=⎧⎪
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⎪⎩(π2−x)(π2−x);0≤x≤π2(π2−x)(π2−(π−x));π2≤x≤π(π2−(2π−x))(π2−(π−x));π≤x<3π2(π2−(2π−x))(π2−(x−2π));3π2≤x≤2π