Consider the function f(x)= cc x^2, & x 1 x+1, & x<1 . Assertion … - Vidyadip

Question

Consider the function $f(x)=\left\{\begin{array}{cc}x^2, & x \geq 1 \\ x+1, & x<1\end{array}\right.$
Assertion (A) : $f$ is not derivable at $x=1$ as $\lim _{x \rightarrow 1^{-}} f(x) \neq \lim _{x \rightarrow 1^{+}} f(x)$.
Reason (R) : If a function $f$ is derivable at a point ' $a$ ', then it is continuous at ' $a$ '.

✓

Answer

(a) : Reason is a standard result.
Also $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1}(x+1)=2$
and $\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1} x^2=1$
$
\Rightarrow \quad \lim _{x \rightarrow 1^{-}} f(x) \neq \lim _{x \rightarrow 1^{+}} f(x)
$
$\Rightarrow f$ is not continuous at $x=1$
$\Rightarrow f$ is not derivable at $x=1$ (From Reason)

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