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Question: If ƒ(x) = \(\frac{a\sin x + b\cos x}{c\sin x + d\cos x}\)decreases for all x if –...

If ƒ(x) = asin⁡x+bcos⁡xcsin⁡x+dcos⁡x\frac{a\sin x + b\cos x}{c\sin x + d\cos x}decreases for all x if –

A

ad – bc < 0

B

ad – bc > 0

C

ab – cd > 0

D

ab – cd < 0

Answer

ad – bc < 0

Explanation

Solution

Ģ(x)

= (acos⁡x−bsin⁡x)(csin⁡x+dcos⁡x)−(asin⁡x+bcos⁡x)(ccos⁡x–dsin⁡x)(csin⁡x+dcos⁡x)2\frac{(a\cos x - b\sin x)(c\sin x + d\cos x) - (a\sin x + b\cos x)(c\cos x–d\sin x)}{(c\sin x + d\cos x)^{2}}

= ad(cos⁡2x+sin⁡2x)−bc(cos⁡2x+sin⁡2x)(csin⁡x+dcos⁡x)2\frac{ad(\cos^{2}x + \sin^{2}x) - bc(\cos^{2}x + \sin^{2}x)}{(c\sin x + d\cos x)^{2}}

= ad−bc(csin⁡x+dcos⁡x)2\frac{ad - bc}{(c\sin x + d\cos x)^{2}}

ƒdecreases for all x if and only if ƒ¢(x) < 0 for all x i.e., ad – bc < 0.