Continuity And Differentiability Question 10
Question 10 - 2024 (31 Jan Shift 2)
Consider the function $f:(0, \infty) \rightarrow R$ 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 $m+n$ is
(1) 0
(2) 3
(3) 1
(4) 2
Show Answer
Answer (4)
Solution
$f:(0, \infty) \rightarrow R$
$f(x)=e^{-\left|\log _0 x\right|}$
$f(x)=\frac{1}{e^{|\ln x|}}=\begin{cases}\frac{1}{e^{-\ln x}} ; 0<x<1 \\ \frac{1}{e^{\ln x}} ; x \geq 1\end{cases}$
$\begin{cases}\frac{1}{\frac{1}{x}}=x ; 0<x<1 \\ \frac{1}{x}, x \geq 1\end{cases}$
$m=0$ (No point at which function is not continuous)
$n=1$ (Not differentiable)
$\therefore m+n=1$
