Consider the function f:(0, ) R defined by f(x)=e^ - | _e x | . I… - Vidyadip

Question

Consider the function $f:(0, \infty) \rightarrow R$ defined by $f(x)=e^{-\left|\log _e x\right|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $\mathrm{m}+\mathrm{n}$ is

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Answer

$f:(0, \infty) \rightarrow R$$ f(x)=e^{-\left|\log _e x\right|}$
$\mathrm{f}(\mathrm{x})=\frac{1}{\mathrm{e}^{\ln x \mid}}=\left\{\begin{array}{l}\frac{1}{\mathrm{e}^{-\ln x}} ; 0 < \mathrm{x} < 1 \\ \frac{1}{\mathrm{e}^{\ln x}} ; \mathrm{x} \geq 1\end{array}\right.$
$\left\{\begin{array}{l}\frac{1}{\frac{1}{x}}=x ; 0 < x < 1 \\ \frac{1}{x}, x \geq 1\end{array}\right.$
${m}=0 \ ($No point at which function is not continuous$)$
${n}=1\ ($Not differentiable$)$
$\therefore \mathrm{m}+\mathrm{n}=1$

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