Which of the following statements is correct for the function

Question:

Which of the following statements is correct for the function $g(\alpha)$ for $\alpha \in R$ such that

$g(\alpha)=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin ^{\alpha} x}{\cos ^{a} x+\sin ^{\alpha} x} d x$

  1. (1) $g(\alpha)$ is a strictly increasing function

     

  2. (2) $\mathrm{g}(\alpha)$ has an inflection point at $\alpha=-\frac{1}{2}$

  3. (3) $\mathrm{g}(\alpha)$ is a strictly decreasing function

  4. (4) $\mathrm{g}(\alpha)$ is an even function


Correct Option: 4,

Solution:

$g(\alpha)=\int_{\frac{\pi}{6}}^{\pi / 3} \frac{\sin ^{\alpha} x}{\left(\sin ^{\alpha} x+\cos ^{a} x\right)} \ldots . .(i)$

$g(\alpha)=\int_{\frac{\pi}{6}}^{\pi / 3} \frac{\cos ^{\alpha} x}{\left(\sin ^{\alpha} x+\cos ^{\circ} x\right)} \ldots(i i)$

$(1)+(2)$

$2 g(\alpha)=\frac{\pi}{6}$

$g(\alpha)=\frac{\pi}{12}$

Constant and even function

Due to typing mistake it must be bonus.

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