Question

If $f\left(x\right)=\{\begin{array}{cc}{4}^x,& -1\le x<1\\ 5-x,& 1\le x\le 5\end{array},$ then

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

Answer 1

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

$\underset{x\to 1^{-}}{lim}f\left(x\right)=\underset{x\to 1^{-}}{lim}{4}^x=4$
$\underset{x\to 1^{+}}{lim}f\left(x\right)=\underset{x\to 1^{+}}{lim}(5-x)=4$
$L.H.L.=R.H.L.=f\left(1\right)$
$\therefore$ It is continuous at $x=1$

$L.H.D.=\underset{h\to 0^{+}}{lim}\frac{f(1-h)-f\left(1\right)}{-h}$
$=\underset{h\to 0^{+}}{lim}\frac{4^{1-h}-4}{-h}=4\ln 4$
$R.H.D.=\underset{h\to 0^{+}}{lim}\frac{f(1+h)-f\left(1\right)}{h}$
$=\underset{h\to 0^{+}}{lim}\frac{5-(1+h)-4}{h}=-1$

$\Rightarrow L.H.D.\ne R.H.D.$
$\therefore$ It is not differentiable at $x=1$