Continuity And Differentiability Question 10
Question 10 - 2024 (31 Jan Shift 2)
Consider the function $\mathrm{f}:(0, \infty) \rightarrow \mathrm{R}$ defined by $f(x)=e^{-\left|\log _{6} x\right|}$. 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
(1) 0
(2) 3
(3) 1
(4) 2
Show Answer
Answer (4)
Solution
$f:(0, \infty) \rightarrow R$
$f(x)=e^{-\left|\log _{0} x\right|}$
$f(x)=\frac{1}{e^{|\ln x|}}=\left{\begin{array}{l}\frac{1}{e^{-\ln x}} ; 0<x<1 \ \frac{1}{e^{\ln x}} ; x \geq 1\end{array}\right.$
$\left{\begin{array}{l}\frac{1}{\frac{1}{x}}=x ; 0<x<1 \ \frac{1}{x}, x \geq 1\end{array}\right.$
$\mathrm{m}=0$ (No point at which function is not continuous)
$\mathrm{n}=1$ (Not differentiable)
$\therefore \mathrm{m}+\mathrm{n}=1$
