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Question

The function asinx+bcosxcsinx+dcosx is decreasing, if


A

ad-bc>0

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B

ad-bc<0

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C

ab-ad>0

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D

ab-cd<0

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Solution

The correct option is B

ad-bc<0


Explanation for the correct option

The given function, fx=asinx+bcosxcsinx+dcosx.

Differentiate the given function with respect to x.

ddxfx=ddxasinx+bcosxcsinx+dcosxf'x=csinx+dcosxddxasinx+bcosx-asinx+bcosxddxcsinx+dcosxcsinx+dcosx2ddx(uv)=vdudx-udvdxv2f'x=csinx+dcosx×acosx-bsinx-asinx+bcosx×ccosx-dsinxcsinx+dcosx2f'x=acsinxcosx-bcsin2x+adcos2x-bdsinxcosx-acsinxcosx+adsin2x-bccos2x+bdsinxcosxcsinx+dcosx2f'x=adcos2x+adsin2x-bcsin2x-bccos2xcsinx+dcosx2f'x=adcos2x+sin2x-bcsin2x+cos2xcsinx+dcosx2f'x=ad×1-bc×1csinx+dcosx2sin2(x)+cos2(x)=1f'x=ad-bccsinx+dcosx2

Now, for decreasing intervals f'(x)<0.

ad-bccsinx+dcosx2<0ad-bc<0×csinx+dcosx2csinx+dcosx20ad-bc<0

So, the function asinx+bcosxcsinx+dcosx is decreasing if ad-bc<0.

Hence, the correct option is (B).


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