JEE PYQ: Application Of Derivatives Question 5
Question 5 - 2021 (17 Mar Shift 1)
Which of the following statements is correct for the function $g(\alpha)$ for $\alpha \in \mathbb{R}$ such that
$$g(\alpha) = \int_{\pi/6}^{\pi/3} \frac{\sin^n x}{\cos^n x + \sin^n x},dx$$
(1) $g(\alpha)$ is a strictly increasing function
(2) $g(\alpha)$ has an inflection point at $\alpha = -\frac{1}{2}$
(3) $g(\alpha)$ is a strictly decreasing function
(4) $g(\alpha)$ is an even function
Show Answer
Answer: (4)
Solution
Using $\int_a^b f(x),dx = \int_a^b f(a+b-x),dx$, we get $2g(\alpha) = \frac{\pi}{6}$, so $g(\alpha) = \frac{\pi}{12}$ is constant, hence an even function. Due to a typing error, this was given as bonus.
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