Question

If $f\left(x\right)=\{\begin{array}{ccc}{3}^x& -1\le x\le 1& \\ 4-x& 1\le x\le 4& \end{array}$ ,then

• A. $f\left(x\right)$ is continuous as well as differentiable at $x=1$
• B. $f\left(x\right)$ is continuous but not differentiable at $x=1$
• C. $f\left(x\right)$ is differentiable but not continuous at $x=1$
• D. none of these

Answer 1

Answer: (B)

$f\left(x\right)$ is continuous but not differentiable at $x=1$

$\underset{x\to 1^{-}}{lim}f\left(x\right)=\underset{x\to 1^{-}}{lim}{3}^x=3$

$\underset{x\to 1^{+}}{lim}f\left(x\right)=\underset{x\to 1^{+}}{lim}(4-x)=3$

LHL $=$RHL

$\therefore$ It is continuous at $x=1$

$f^{\prime }\left(x\right)=\{\begin{array}{cc}{3}^x\log 3& -1\le x\le 1\\ -1& 1\le x\le 4\end{array}$

$\underset{x\to 1^{-}}{lim}{f}^{\prime }\left(x\right)=\underset{x\to 1^{-}}{lim}{3}^x\log 3=3\log 3$

$\underset{x\to 1^{+}}{lim}{f}^{\prime }\left(x\right)=\underset{x\to 1^{+}}{lim}(-1)=-1$

LHL$\ne$RHL

$\therefore$ It is not differentiable at $x=1$