JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

$g(x) = f(x) + f(1-x)$ and $f''(x) > 0$ . If $g$ decreases in $(0, \alpha)$ and increases in $(\alpha, 1)$ , find $\tan^{-1}(2\alpha) + \tan^{-1}(\frac{\alpha+1}{\alpha})$ .

Choose the correct answer:

Explanation

Solution:

$g'(x) = f'(x) - f'(1-x)$ .

Critical point ke liye $g'(x) = 0 \implies f'(x) = f'(1-x)$ .

Kyunki $f''(x) > 0$ , $f'(x)$ ek monotonic function hai, isliye $x = 1-x \implies x = 1/2$ .

Toh $\alpha = 1/2$ .

Value find karni hai: $\tan^{-1}(2 \cdot \frac{1}{2}) + \tan^{-1}(\frac{1/2+1}{1/2})$

= \tan^{-1}(1) + \tan^{-1}(3) = \frac{\pi}{4} + \tan^{-1}(3)

Using $\tan^{-1} x + \tan^{-1} y = \pi + \tan^{-1}(\frac{x+y}{1-xy})$ (kyunki $xy > 1$ ):

= \pi + \tan^{-1}(\frac{1+3}{1-3}) = \pi + \tan^{-1}(-2)

Sahi Option: (3) $\tan^{-1}(-2)$