The function frac{asin x + bcos x}{csin x + d | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $y = \frac{a\sin x + b\cos x}{c\sin x + d\cos x}$.The function is decreasing when $\frac{dy}{dx} < 0$.Using the quotient rule,$\frac{dy}{dx} =

Vedclass · Accepted Answer

$ad - bc < 0$

Vedclass · Answer

(A) Let $y = \frac{a\sin x + b\cos x}{c\sin x + d\cos x}$.The function is decreasing when $\frac{dy}{dx} Using the quotient rule,$\frac{dy}{dx} = \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}$.Expanding the numerator:$= (ac\sin x \cos x - bc\sin^2 x + ad\cos^2 x - bd\sin x \cos x) - (ac\sin x \cos x - ad\sin^2 x + bc\cos^2 x - bd\sin x \cos x)$.Simplifying the expression:$= ad(\cos^2 x + \sin^2 x) - bc(\sin^2 x + \cos^2 x) = ad - bc$.Since the denominator $(c\sin x + d\cos x)^2$ is always positive,the condition $\frac{dy}{dx} < 0$ implies $ad - bc < 0$.

The function $\frac{a\sin x + b\cos x}{c\sin x + d\cos x}$ is decreasing,if

Similar Questions

If the function $f(x) = \cos |x| - 2ax + b$ is an increasing function on the entire real line,then what is the value of $a$?

Find the intervals in which the function $f(x) = -2x^{3} - 9x^{2} - 12x + 1$ is strictly increasing or strictly decreasing.

The function $f(x) = \tan^{-1}(\sin x + \cos x)$,$x > 0$ is always an increasing function on the interval

The function $f(x) = \sin^4 x + \cos^4 x$ increases if

The function $f(x) = x + \cos x$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module