Question

The function $\frac{\sin (x+\alpha )}{\sin (x+\beta )}$ has no maximum or minimum if (k an integer)

• A. $\beta -\alpha =k\pi$
• B. $\beta -\alpha \ne k\pi$
• C. $\beta -\alpha =2k\pi$
• D. none of the above

Answer 1

The correct option is B $\beta -\alpha \ne k\pi$

$f\left(x\right)=\frac{sin(x+\alpha )}{sin(x+\beta )}$

$f^{\prime }\left(x\right)=\frac{sin(x+\beta ).cos(x+\alpha )-sin(x+\alpha ).cos(x+\beta )}{si{n}^2(x+\beta )}=-\frac{sin(\alpha -\beta )}{si{n}^2(x+\beta )}$
So for no minima or maxima
$f^{\prime }\left(x\right)\ne 0\Rightarrow \alpha -\beta \ne k\pi$, where k is any integer