Application of Derivatives · Mathematics · IAT (IISER)

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a strictly decreasing function with $|f(t)|<\pi / 2$ for all $t \in \mathbf{R}$. Let $g:[0, \pi] \rightarrow$ R be a function defined by $g(t)=\sin (f(t))$. Which one of the following statements is Correct?

What is the largest area of a rectangle, whose sides are parallel to the coordinate axes, that can be inscribed under the graph of the curve $y=1-x^2$ and above the $x$-axis?

Let $\alpha$ be a real number. What is the total number of distinct point(s) of intersection between the parabola $y=x^2+4 x \sin \alpha+6$ and the pair of lines $y^2=1$ ?

Let $f(x)=\sin (3 x), x \in\left[0, \frac{\pi}{2}\right]$. Which of the following statements is true