Question

Prove that the function f (x) = sinx is
(i) strictly increasing in $open parentheses 0 comma space straight pi over 2 close parentheses$
(ii) strictly decreasing in $open parentheses straight pi over 2 comma space straight pi close parentheses$
(iii) neither increasing nor decreasing in $left parenthesis 0 comma space straight pi right parenthesis$ .

Solution

Here $straight f left parenthesis straight x right parenthesis space equals space sinx$
$therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cos space straight x$
(i) Since for each $straight x space element of space space space open parentheses 0 comma space space straight pi over 2 close parentheses comma space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cos space straight x thin space greater than 0$
$therefore space space space space space straight f space space is space strictly space increasing space in space space open parentheses 0 comma space space straight pi over 2 close parentheses$
(ii) Since for each $straight x space element of space space open parentheses straight pi over 2 comma space straight pi close parentheses comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cos space straight x thin space less than space 0$
$therefore space space space space space straight f space space is space strictly space decreasing space in space open parentheses straight pi over 2 comma space space straight pi close parentheses.$
(iii) f(x) is strictly increasing in $open parentheses 0 comma space straight pi over 2 close parentheses$ and strictly decreasing in $open parentheses straight pi over 2 comma space straight pi close parentheses$
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Application Of Derivatives

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