The function f(x) = tan^-1 (sinx + cosx) is an increasing function in

(π/4, π /2)

(–π/2, π /4)

(0, π /2)

(–π/2, π /2)

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Solution

(π/4, π /2)

straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator 1 plus space left parenthesis sin space straight x space plus cos space straight x right parenthesis squared end fraction left parenthesis cos space straight x minus space sin space straight x right parenthesis space equals space fraction numerator square root of 2 cos space open parentheses straight x space plus space begin display style straight pi over 4 end style close parentheses over denominator 1 plus space left parenthesis sin space straight x space plus space cos space straight x right parenthesis squared end fraction straight f left parenthesis straight x right parenthesis space is space increasing space if space minus space straight pi over 2 space less than thin space straight x space plus space straight pi over 4 space less than thin space straight pi over 2 minus fraction numerator 3 straight pi over denominator 4 end fraction space less than thin space straight x space less than thin space straight pi over 4 hence space fI left parenthesis straight x right parenthesis space is space increasing space when space straight x element of open parentheses fraction numerator negative straight pi over denominator 2 end fraction comma straight pi over 4 close parentheses

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