$A$ 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 $\text{ m}$)

Let $p(x)$ be a polynomial of degree $4$ having extrema at $x=1$ and $x=2$,and $\lim_{x \rightarrow 0} \left(1+\frac{p(x)}{x^2}\right) = 2$. Then the value of $p(2)$ is

The coordinates of the point $P$ on the graph of the function $y = e^{-|x|}$ where the portion of the tangent intercepted between the coordinate axes has the greatest area,is

In the interval $[-2, 4]$,the absolute maximum of $f(x) = 2x^3 - 3x^2 - 12x + 5$ occurs at $x =$

$x$ and $y$ are two variables such that $x > 0$ and $xy = 1$. Then the minimum value of $x + y$ is

Let $f(x) = x \sin \pi x$,$x > 0$. Then for all natural numbers $n$,$f^{\prime}(x)$ vanishes at
$(A)$ a unique point in the interval $\left(n, n+\frac{1}{2}\right)$
$(B)$ a unique point in the interval $\left(n+\frac{1}{2}, n+1\right)$
$(C)$ a unique point in the interval $(n, n+1)$
$(D)$ two points in the interval $(n, n+1)$