$\ell n f(x)=x^{2}(\ell \ln 2-\ell n x)$
$f^{\prime}(x)=f(x)\{-x+(\ell n 2-\ell n x) 2 x\}$
$f^{\prime}(x)=\underbrace{f(x)}_{+} \cdot \underbrace{x}_{+} \underbrace{(2 \ell n 2-2 \ell n x-1)}_{g(x)}$
$g(x)=2 \ell n^{2}-2 \ell n x-1$
$=\ell n \frac{4}{x^{2}}-1=0 \Rightarrow x=\frac{2}{\sqrt{e}}$
$\mathrm{LM}=\frac{2}{\sqrt{\mathrm{e}}}$
Local maximum value $=\left(\frac{2}{2 / \sqrt{\mathrm{e}}}\right)^{\frac{4}{\mathrm{e}}} \Rightarrow \mathrm{e}^{\frac{2}{\mathrm{e}}}$
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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 $3.$