# Which of the following statements

Question:

Which of the following statements is incorrect for the function $\mathrm{g}(\alpha)$ for $\alpha \in \mathrm{R}$ such that.

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

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

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

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

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

Correct Option: , 4

Solution:

$\mathrm{g}(\alpha)=\int_{\frac{\pi}{6}}^{\pi / 3} \frac{\sin ^{\alpha} \mathrm{x}}{\left(\sin ^{\alpha} \mathrm{x}+\cos ^{\alpha} \mathrm{x}\right)} \quad \ldots . .(\mathrm{i})$...(1)

$g(\alpha)=\int_{\frac{\pi}{6}}^{\pi / 3} \frac{\cos ^{\alpha} x}{\left(\sin ^{\alpha} x+\cos ^{\alpha} x\right)} \ldots \ldots$..(2)

$(1)+(2)$

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

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

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