Application of Derivatives

Which of the following function are strictly decreasing on (a) cos x (b) cos 2x (c) cos 3x (d) tan x - Wired Faculty

Question

Which of the following function are strictly decreasing on $open parentheses 0 comma space straight pi over 2 close parentheses space ?$
(a) cos x (b) cos 2x (c) cos 3x (d) tan x

Solution

(a) Let $straight f left parenthesis straight x right parenthesis space equals space cos space straight x comma space space space space space space space space therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus sin space straight x space$
In $open parentheses 0 comma space straight pi over 2 close parentheses comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus sin space straight x space less than space 0$
$therefore space space space space straight f left parenthesis straight x right parenthesis space equals space cosx space is space strictly space decreasing space on space open parentheses 0 comma space straight pi over 2 close parentheses$
(b) Let $straight f left parenthesis straight x right parenthesis space equals space cos space 2 straight x comma space space space space space space therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 2 space sin space 2 straight x$
In $open parentheses 0 comma space straight pi over 2 close parentheses comma space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 2 space sin space 2 straight x space less than space 0$
$therefore space space space space space straight f left parenthesis straight x right parenthesis space equals space cos space 2 straight x$ is strictly decreasing on $open parentheses 0 comma space straight pi over 2 close parentheses.$
(c) Let f(x) = cos 3x, $therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus 3 space sin space 3 straight x$
$0 less than straight x less than straight pi over 2$
$rightwards double arrow space space space space space space 0 less than space 3 straight x space less than space fraction numerator 3 straight pi over denominator 2 end fraction$
Now $straight f apostrophe left parenthesis straight x right parenthesis space less than 0 space space space space when space 0 space less than space 3 straight x space less than space straight pi$
and $straight f apostrophe left parenthesis straight x right parenthesis space greater than 0 space space space when space straight pi space less than space 3 straight x space less than fraction numerator 3 straight pi over denominator 2 end fraction$
$therefore space space space space space straight f left parenthesis straight x right parenthesis space is space not space straight a space strictly space decreasing space function space on space open parentheses 0 comma space space straight pi over 2 close parentheses$
(d) Let $straight f left parenthesis straight x right parenthesis space equals tanx comma space space space space space space space space therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space sec squared straight x$
Now, $straight f apostrophe left parenthesis straight x right parenthesis greater than 0 space space for all space straight x space space space space element of space space space open parentheses 0 comma space space straight pi over 2 close parentheses$
$therefore space space space straight f left parenthesis straight x right parenthesis space is space not space straight a space strictly space decreasing space function space on space open parentheses 0 comma space space space straight pi over 2 close parentheses$

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Application Of Derivatives

Which of the following function are strictly decreasing on $open parentheses 0 comma space straight pi over 2 close parentheses space ?$
(a) cos x (b) cos 2x (c) cos 3x (d) tan x

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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