Question

Which of the following statements is incorrect for the function $g\left(\alpha \right)$ for $\alpha \in R$ such that $g\left(\alpha \right)=\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}\frac{{\sin }^{\alpha }x}{{\cos }^{\alpha }x+{\sin }^{\alpha }x}dx$

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

Answer 1

The correct option is C $g\left(\alpha \right)$ is an even function

Given : $g\left(\alpha \right)=\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}\frac{{\sin }^{\alpha }x}{{\cos }^{\alpha }x+{\sin }^{\alpha }x}dx$
Using $x\to \frac{\pi }{2}-x$
$\Rightarrow g\left(\alpha \right)=\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}\frac{{\cos }^{\alpha }x}{{\sin }^{\alpha }x+{\cos }^{\alpha }x}dx$
Adding both, we get
$\Rightarrow 2g\left(\alpha \right)=\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}dx\Rightarrow g\left(\alpha \right)=\frac{\pi }{12}$

Note: In question it is asked incorrect, then A, B, D are incorrect so this should be BONUS marks

Although if correct was asked then only C satisfies.