Question

Assertion :The function $f\left(x\right)=\{\begin{array}{cc}\left|x\right|+3& \forall x\le -3\\ -2x& \forall -3<x<3\\ 6x+2& \forall x\ge 3\end{array}$ is continuous for $-3\le x<3$ & $x>3$ & non differentiable at $x=3$. Reason: If ${lim}_{x\to a}f\left(x\right)=f\left(a\right)$ then $f\left(x\right)$ is continuous and if L.H.D at $x=a$=R.H.D at $x=a$ then $f\left(x\right)$ is non differentiable at $x=0$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

At $x=-3$, $f(-3)=6$
and $\underset{x\to 3^{-}}{lim}f\left(x\right)$ $\Rightarrow$ $\underset{x\to 3^{-}}{lim}\left|x\right|+3=6=\underset{x\to 3^{+}}{lim}\left|x\right|+3$
$\therefore$ R.H.L.=L.H.L.=$f(-3)=6$
$\therefore$ f(x) is continuous at $x=-3$
Again, at $x=3$
L.H.L.$=\underset{x\to 3^{-}}{lim}(-2x)=-6$
and R.H.L.$=\underset{x\to 3^{+}}{lim}f\left(x\right)=\underset{x\to 3^{+}}{lim}6x+2=20$
$\therefore$ L.H.L. of f(x) at x=3 $\ne$ R.H.L. of f(x) at $x=3$
$\Rightarrow$ f(x) is not continuous at $x=3$
$\Rightarrow$ f(x) can not be differentiable at $x=3$
Also at $x=x_0$ where $-3<x_0<3$, $\underset{x\to x_0}{lim}f\left(x\right)=f\left(x_0\right)$
so f(x) is continuous for $-3<x<3$