Prove that the function log sin x is strictly increasing on $open parentheses 0 comma space space straight pi over 2 close parentheses$ and strictly decreasing in $open parentheses straight pi over 2 comma space space straight pi close parentheses.$

Solution

Let $straight f left parenthesis straight x right parenthesis space equals space log space sin space straight x space$
$therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator sin space straight x end fraction. space cos space straight x space equals space cot space straight x$
(i) Now, $straight f apostrophe left parenthesis straight x right parenthesis space equals space cot space straight x space greater than space 0 space space space space for space space space straight x space element of space open parentheses 0 comma space space straight pi over 2 close parentheses$
$therefore space space space space space straight f left parenthesis straight x right parenthesis space is space increasing space in space open parentheses 0 comma space space space straight pi over 2 close parentheses$
$left parenthesis ii right parenthesis space straight f apostrophe left parenthesis straight x right parenthesis space equals space cot space straight x space less than 0 space space space for space space space space straight x space element of space space open parentheses straight pi over 2 comma space straight pi close parentheses therefore space space space space space straight f left parenthesis straight x right parenthesis space space is space decreasing space in space open parentheses straight pi over 2 comma space straight pi close parentheses.$

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Application Of Derivatives

Prove that the function log sin x is strictly increasing on $open parentheses 0 comma space space straight pi over 2 close parentheses$ and strictly decreasing in $open parentheses straight pi over 2 comma space space straight pi close parentheses.$

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

Prove that the function log sin x is strictly increasing on and strictly decreasing in

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Prove that the function log sin x is strictly increasing on $open parentheses 0 comma space space straight pi over 2 close parentheses$ and strictly decreasing in $open parentheses straight pi over 2 comma space space straight pi close parentheses.$