Question

If the function

cos x+k sin x

is

2 cos x+3 sin x decreasing in its domain then

K belong to​

Answer 1

SOLUTION

TO DETERMINE

The range of k for which the below function is decreasing in its domain

$\displaystyle \sf{f(x) = \frac{ \cos x + k \: sin x}{2 \cos x + 3 \sin x} }$

CONCEPT TO BE IMPLEMENTED

A function f(x) defined in a domain D is said to be decreasing in the domain if

f'(x) < 0 for every x ∈ D

EVALUATION

Here the given function is

$\displaystyle \sf{f(x) = \frac{ \cos x + k \: sin x}{2 \cos x + 3 \sin x} }$

Differentiating both sides with respect to x we get

$\displaystyle \sf{f'(x) = \frac{(2 \cos x + 3 \sin x) \frac{d}{dx} ( \cos x + k \sin x) - ( \cos x + k \sin x) \frac{d}{dx} (2 \cos x + 3 \sin x)}{ {(2 \cos x + 3 \sin x)}^{2} } }$

$\implies\displaystyle \sf{f'(x) = \frac{(2 \cos x + 3 \sin x) ( - \sin x + k \cos x) - ( \cos x + k \sin x) ( - 2 \sin x + 3 \cos x)}{ {(2 \cos x + 3 \sin x)}^{2} } }$

$\implies \displaystyle \sf{f'(x) = \frac{ - 2 \cos x \sin x + 2k { \cos}^{2}x - 3 { \sin}^{2}x + 3k\cos x \sin x + 2 \cos x \sin x - 3 { \cos}^{2}x + 2k { \sin}^{2}x - 3k \cos x \sin x }{ {(2 \cos x + 3 \sin x)}^{2} } }$

$\implies \displaystyle \sf{f'(x) = \frac{ 2k { \cos}^{2}x - 3 { \sin}^{2}x - 3 { \cos}^{2}x + 2k { \sin}^{2}x }{ {(2 \cos x + 3 \sin x)}^{2} } }$

$\implies \displaystyle \sf{f'(x) = \frac{ 2k( { \cos}^{2}x + { \sin}^{2} x)- 3 ({ \sin}^{2}x + { \cos}^{2}x ) }{ {(2 \cos x + 3 \sin x)}^{2} } }$

$\implies \displaystyle \sf{f'(x) = \frac{ 2k - 3 }{ {(2 \cos x + 3 \sin x)}^{2} } }$

Now f(x) is decreasing

$\therefore \: \: \displaystyle \sf{f'(x) < 0 }$

$\implies \displaystyle \sf{ \frac{ 2k - 3}{ {(2 \cos x + 3 \sin x)}^{2} } < 0 }$

$\implies \displaystyle \sf{ 2k - 3 < 0} \: ( \: \because \: denominator \: is \: positive)$

$\implies \displaystyle \sf{ 2k < 3}$

$\implies \displaystyle \sf{ k < \frac{3}{2} }$

$\implies \displaystyle \sf{k \: \in \: \bigg( - \infty \:, \: \frac{3}{2} \: \bigg) }$

FINAL ANSWER

$\boxed{ \: \: \displaystyle \sf{k \: \in \: \bigg( - \infty \:, \: \frac{3}{2} \: \bigg) } \: }$

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