The function y = e^{-|x|} is | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) We have,$f(x) = \begin{cases} e^x, & x < 0 \ e^{-x}, & x \ge 0 \end{cases}$Clearly,$f(x)$ is continuous and differentiable for all $x 
eq 0$.Now

Vedclass · Accepted Answer

Continuous but not differentiable at $x = 0$

Vedclass · Answer

(C) We have,$f(x) = \begin{cases} e^x, & x Clearly,$f(x)$ is continuous and differentiable for all $x 
eq 0$.Now,$\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0} e^x = 1$ and $\lim_{x 	o 0^+} f(x) = \lim_{x 	o 0^+} e^{-x} = 1$.Also,$f(0) = e^0 = 1$. Since $\lim_{x 	o 0^-} f(x) = \lim_{x 	o 0^+} f(x) = f(0)$,the function is continuous at $x = 0$.Now,check for differentiability at $x = 0$:Left-hand derivative $(LHD)$ at $x = 0$ is $\frac{d}{dx}(e^x) |_{x=0} = e^0 = 1$.Right-hand derivative $(RHD)$ at $x = 0$ is $\frac{d}{dx}(e^{-x}) |_{x=0} = -e^0 = -1$.Since $LHD 
eq RHD$ at $x = 0$,the function is not differentiable at $x = 0$.Thus,$f(x) = e^{-|x|}$ is continuous but not differentiable at $x = 0$.

The function $y = e^{-|x|}$ is

Similar Questions

Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x) - f(y)| \le 2|x - y|^{\frac{3}{2}}$ for all $x, y \in R$. If $f(0) = 1$,then $\int_{0}^{1} f^2(x) dx$ is equal to

Let $R$ denote the set of all real numbers. Define the function $f: R \rightarrow R$ by $f(x) = \begin{cases} 2-2x^2-x^2 \sin \frac{1}{x} & \text{if } x \neq 0 \\ 2 & \text{if } x=0 \end{cases}$. Then which one of the following statements is True?

If $f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 1 \\ ax^2 + b, & -1 < x < 1 \end{cases}$ is differentiable $\forall x \in \mathbb{R}$,then one of the values of $a$ and $b$ is-

If the function $g(x)=\begin{cases} K \sqrt{x+1} &, 0 \leq x \leq 3 \\ mx+2 &, 3 < x \leq 5 \end{cases}$ is differentiable,then $K+m=$

The domain of the derivative of the function $f(x) = \begin{cases} \tan^{-1}x, & |x| \le 1 \\ \frac{1}{2}(|x| - 1), & |x| > 1 \end{cases}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module