If {f}(x)=a x+b x/c x+d x is decreasing for all x then

Correct Answer is : ad−bc<0a d-b c < 0ad−bc<0

Correct Answer is : ad−bc<0a d-b c < 0ad−bc<0

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MHTCET 16 May 2024 Shift 2 Paper

Question : 120 of 150

Marks: +1, -0

\mathrm{f}(x)=\frac{a \sin x+b \cos x}{c \sin x+d \cos x}

x

then

Solution:

Concept:

A function is decreasing when its derivative is negative for all

x

Explanation:

Given

f(x)=\frac{a \sin x+b \cos x}{c \sin x+d \cos x}

Differentiate using the quotient rule:

f'(x)=\frac{(c\sin x+d\cos x)(a\cos x-b\sin x)-(a\sin x+b\cos x)(c\cos x-d\sin x)}{(c\sin x+d\cos x)^2}

Expanding the numerator, the

ac\sin x\cos x

and

bd\sin x\cos x

terms cancel out.

The remaining terms give:

ad(\sin^2 x+\cos^2 x)-bc(\sin^2 x+\cos^2 x)=ad-bc

Thus,

f'(x)=\frac{ad-bc}{(c\sin x+d\cos x)^2}

Since the denominator is always positive,

f'(x) < 0

implies

ad-bc < 0

Answer:

ad-bc < 0

, i.e. option B.

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