Question

Which of the following statements is incorrect for the function $g(\alpha)$ for $\alpha \in R$ such that $g(\alpha)=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin ^{\alpha} x}{\cos ^{\alpha} x+\sin ^{\alpha} x} d x$

• A. $g (\alpha)$ is a strictly increasing function
• B. $g (\alpha)$ has an inflection point at $\alpha=-\frac{1}{2}$
• C. $g (\alpha)$ is a strictly decreasing function
• D. $g (\alpha)$ is an even function

Answer 1

Answer: (D)

$
\begin{array}{l}
g(\alpha)=\int_{\frac{\pi}{6}}^{\pi / 3} \frac{\sin ^{\alpha} x}{\left(\sin ^{\alpha} x+\cos ^{\alpha} x\right)} \ldots \text { (i) } \\
g(\alpha)=\int_{\frac{\pi}{6}}^{\pi / 3} \frac{\cos ^{\alpha} x}{\left(\sin ^{\alpha} x+\cos ^{\alpha} x\right)} \ldots \text { (ii) }
\end{array}
$
$(1)+(2)$
$
2 g(\alpha)=\frac{\pi}{6}
$
$
g(\alpha)=\frac{\pi}{12}
$
Constant and even function
Due to typing mistake it must be bonus.