Application of Derivatives

Question

Show that f (x) = tan ^-¹ (sin x + cos x) is a strictly increasing function in the interval $open parentheses 0 comma space space straight pi over 4 close parentheses$

Solution

Here $straight f left parenthesis straight x right parenthesis space equals space tan to the power of negative 1 end exponent left parenthesis sinx plus cosx right parenthesis$
$therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator 1 plus left parenthesis sin space straight x plus space cos space straight x right parenthesis squared end fraction. space straight d over dx left parenthesis sin space straight x space plus space cos space straight x right parenthesis$
$equals space fraction numerator 1 over denominator 1 plus sin squared straight x plus cos squared straight x plus 2 sinx space cosx space end fraction. space left parenthesis cosx minus sinx right parenthesis space equals space fraction numerator cos space straight x minus space sinx space over denominator 1 plus 1 space plus space sin space 2 straight x end fraction equals space minus fraction numerator cosx minus sin space straight x over denominator 2 plus sin space 2 straight x end fraction greater than 0 space space space for all space space straight x space space space element of space space open parentheses 0 comma space space straight pi over 4 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space cos space straight x space greater than space sin space straight x space space where space straight x space element of space open parentheses 0 comma space straight pi over 4 close parentheses close square brackets space therefore space space space space straight f left parenthesis straight x right parenthesis space is space increasing space in space open parentheses 0 comma space space straight pi over 4 close parentheses. space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space$

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