Application of Derivatives

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Prove that the function f given by f(x) = log cos x is strictly decreasing on and strictly increasing on - Wired Faculty

Question

Prove that the function f given by f(x) = log cos x is strictly decreasing on $open parentheses 0 comma space straight pi over 2 close parentheses$ and strictly increasing on $open parentheses straight pi over 2 comma space straight pi close parentheses.$

Solution

straight f left parenthesis straight x right parenthesis space equals space log space cos space straight x

therefore space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals 1 over cos. left parenthesis negative sin space straight x right parenthesis space equals space minus tan space straight x

(i) Now,

straight f apostrophe left parenthesis straight x right parenthesis space equals space minus tan space straight x space less than space 0 space space for space space straight x space element of space open parentheses 0 comma space space straight pi over 2 close parentheses

therefore space space space straight f space is space strictly space decreasing space on space open parentheses 0 comma space space straight pi over 2 close parentheses

left parenthesis ii right parenthesis space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus tan space straight x greater than 0 space space space for space space straight x space element of space open parentheses straight pi over 2 comma space straight pi close parentheses therefore space space space space space straight f space is space strictly space increasing space on space open parentheses straight pi over 2 comma space space straight pi close parentheses.