The maximum value of $f(x) = (x + 1)^{\frac{1}{3}} - (x - 1)^{\frac{1}{3}}$ for $x \in [0, 1]$ is ....

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 maximum slope of the curve $y = \frac{1}{2} x^{4} - 5 x^{3} + 18 x^{2} - 19 x$ occurs at the point

The least value of the sum of any positive real number and its reciprocal is

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in $sq.m$) of the flowerbed is

For a given perimeter,the triangle having the maximum area is: