Question

The function $\frac{[\sin (x+\alpha \left)\right]}{[\sin (x+\beta \left)\right]}$ has no maximum or minimum if

• A. $\beta –ɑ=K\pi$
• B. $\beta –ɑ\ne K\pi$
• C. $\beta –ɑ=2K\pi$
• D. $Noneofthese$

Answer 1

The correct option is B

$\beta –ɑ\ne K\pi$

Explanation for correct option

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

$\Rightarrow f\text{'}\left(x\right)=\frac{\sin (x+\beta )·\cos (x+\alpha )–\sin (x+\alpha )·\cos (x+\beta )}{{\sin }^2(x+\beta )}$

On expanding each and every term we get,

$\Rightarrow f\text{'}\left(x\right)=\frac{\sin (\beta -\alpha )}{{\sin }^2(x+\beta )}$

For no maxima and minima,

$\Rightarrow f\text{'}\left(x\right)\ne 0$

$\Rightarrow \frac{\sin (\beta -\alpha )}{{\sin }^2(x+\beta )}\ne 0$

So, from the above equation

$\sin (\beta -\alpha )\ne 0$

$\Rightarrow \beta -\alpha \ne K\mathrm{\pi }$ [where, $K=0,1,2,3,.......$ ]

Hence, option B is correct.