List of top Mathematics Questions on Applications of Derivatives asked in JEE Main

Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) + 3f\left( \frac{\pi}{2} - x \right) = \sin x, x \in \mathbb{R}$. Let the maximum value of $f$ on $\mathbb{R}$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^2$ and $h(x) = \beta x^3, \beta>0$, is $\alpha^2$, then $30\beta^3$ is equal to :}

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( f''(x)>0 \) for all \( x \in \mathbb{R} \) and \( f'(a-1) = 0 \), where \( a \) is a real number. Let \( g(x) = f(\tan^2 x - 2\tan x + a) \), \( 0<x<\frac{\pi}{2} \).
Consider the following two statements :
(I) \( g \) is increasing in \( (0, \frac{\pi}{4}) \)
(II) \( g \) is decreasing in \( (\frac{\pi}{4}, \frac{\pi}{2}) \)
Then,