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Statement$-1$ If $f R \rightarrow R$ and $c \in R$ is such that $f$ is increasing in $(c - \delta , c)$ and $f$ is decreasing in $(c, c + \delta )$ then $f$ has a local maximum at $c$. Where $\delta$ is a sufficiently small positive quantity.
Statement $-2$ Let $f (a, b) \rightarrow \,R, c \in (a, b)$. Then $f$ can not have both a local maximum and a point of inflection at $x = c.$
Statement $-3 $ The function $f (x) = x^2 | x |$ is twice differentiable at $x = 0.$
Statement $-4$ Let $f [c - 1, c + 1] \rightarrow [a, b]$ be bijective map such that $f$ is differentiable at $c$ then $f^{-1}$ is also differentiable at $f (c)$.
Consider $f(x)=k e^x-x$ for all real $x$ where $k$ is a real constant.
$1.$ The line $\mathrm{y}=\mathrm{x}$ meets $\mathrm{y}=k e^{\mathrm{x}}$ for $\mathrm{k} \leq 0$ at
$(A)$ no point $(B)$ one point
$(C)$ two points $(D)$ more than two points
$2.$ The positive value of $\mathrm{k}$ for which $\mathrm{ke}^{\mathrm{x}}-\mathrm{x}=0$ has only one root is
$(A)$ $1 / \mathrm{e}$ $(B)$ $1$ $(C)$ 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
$(A)$ $\left(0, \frac{1}{\mathrm{e}}\right)$ $(B)$ $\left(\frac{1}{\mathrm{e}}, 1\right)$ $(C)$ $\left(\frac{1}{e}, \infty\right)$ $(D)$ $(0,1)$
Give the answer question $1,2$ and $3.$