Local maximum value of the function $\frac{\log x}{x}$ is

Let $f(x) = \int\limits_0^x \frac{\sin t}{t} dt$ for $x > 0$. Then $f(x)$ has:

What is the minimum value of the sum of a real number $x$ and its reciprocal?

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 $3$.

Let $f(x)=(x-2)^{17}(x+5)^{24}$. Then

Suppose the cubic $x^3 - px + q$ has three distinct real roots where $p > 0$ and $q > 0$. Then which one of the following holds?