The sum of two nonzero numbers is $4$. The minimum value of the sum of their reciprocals is

$A$ box is to be made with a square base and an open top. If the area of the material used is $48 \, m^2$,then the maximum volume of the box is ........... $m^3$.

Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is $\sin ^{-1}\left(\frac{1}{3}\right)$.

If $y = a \log x + bx^2 + x$ has extreme values at $x = -1$ and $x = 2$,then:

If a continuous function $f$ defined on the real line $R$ assumes positive and negative values in $R$,then the equation $f(x)=0$ has a root in $R$. For example,if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum value is negative,then the equation $f(x)=0$ has a root in $R$.
Consider $f(x)=k e^x-x$ for all real $x$,where $k$ is a real constant.
$1.$ The line $y=x$ meets $y=k e^x$ for $k \leq 0$ at
$(A)$ no point $(B)$ one point $(C)$ two points $(D)$ more than two points
$2.$ The positive value of $k$ for which $k e^x-x=0$ has only one root is
$(A)$ $1/e$ $(B)$ $1$ $(C)$ $e$ $(D)$ $\log_e 2$
$3.$ For $k>0$,the set of all values of $k$ for which $k e^x-x=0$ has two distinct roots is
$(A)$ $(0, 1/e)$ $(B)$ $(1/e, 1)$ $(C)$ $(1/e, \infty)$ $(D)$ $(0, 1)$
Give the answer for questions $1, 2$ and $3$.

The function $f(x) = 2x^3 - 3x^2 - 12x + 4, x \in R$ has