Question

If $\alpha, \beta \in\left(\frac{\pi}{2}, \pi\right)$ and $\alpha<\beta$, then which one of the following is true?

• A. $e ^{\cos \alpha-\cos \beta}<\frac{\alpha}{\beta}$
• B. $e ^{\cos \beta-\cos \alpha}<\frac{\beta}{\alpha}$
• C. $e ^{\cos \alpha-\cos \beta}<\frac{\beta}{\alpha}$
• D. $e ^{\cos \beta-\cos \alpha}<\frac{\alpha}{\beta}$

Answer 1

Answer: (B)

Consider $f(x)=\frac{e^{\cos x}}{x}$
Now, $f^{\prime}(x)=\frac{-e^{\cos x}(1+x \sin x)}{x^2}$
As $ f^{\prime}(x)<0 \Rightarrow f(\alpha)>f(\beta)$
So, $ \frac{ e ^{\cos \alpha}}{\alpha}>\frac{ e ^{\cos \beta}}{\beta} \Rightarrow e ^{\cos \alpha-\cos \beta}>\frac{\alpha}{\beta}$.