Which of the following statements is incorrect for the function g… - Vidyadip

MCQ

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
✓
$g (\alpha)$ is an even function

✓

Answer

Correct option: D.

$g (\alpha)$ is an even function

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

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

$(1)+(2)$

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

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

Constant and even function

Due to typing mistake it must be bonus.

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