The ordinates of the points on the curve $y = \tan^{-1}(\sin \sqrt{x})$,$0 \leq x \leq 8\pi^2$,at which the tangent is parallel to the $x$-axis are

Let $f:[0,1] \rightarrow \mathbb{R}$ be a function. Suppose $f$ is twice differentiable,$f(0)=f(1)=0$ and satisfies $f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^x$ for $x \in[0,1]$.
$1.$ Which of the following is true for $0 < x < 1$?
$(A)$ $0 < f(x) < \infty$
$(B)$ $-\frac{1}{2} < f(x) < \frac{1}{2}$
$(C)$ $-\frac{1}{4} < f(x) < 1$
$(D)$ $-\infty < f(x) < 0$
$2.$ If the function $g(x) = e^{-x} f(x)$ assumes its minimum in the interval $[0,1]$ at $x=\frac{1}{4}$,which of the following is true?
$(A)$ $f^{\prime}(x) < f(x)$ for $x \in (0, 1/4)$
$(B)$ $f^{\prime}(x) > f(x)$ for $x \in (0, 1/4)$
$(C)$ $f^{\prime}(x) < f(x)$ for $x \in (1/4, 1)$
$(D)$ $f^{\prime}(x) > f(x)$ for $x \in (1/4, 1)$

If $x - 2y = 4,$ the minimum value of $xy$ is

$A$ missile is fired from the ground level and rises $x$ meters vertically upwards in $t$ seconds, where $x = 100t - \frac{25}{2}t^2$. The maximum height reached is: (in $\text{ m}$)

Show that the function given by $f(x) = \frac{\log x}{x}$ has a maximum at $x = e$.

The maximum area of a right-angled triangle with hypotenuse $h$ is