Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
0 votes

Examine the differentiability of f,where f is defined by $f(x)=\left \{\begin{array}{1 1}1+x, & if\;x\leq 2\\5-x, & if\;x>2\end{array}\right.$at $x=2$.

Can you answer this question?

1 Answer

0 votes
  • A function is not differentiable if LHL $\neq$ RHL.
  • A function is not differentiable if LHL or RHL does not exist.
Step 1:
$f(x)=\left \{\begin{array}{1 1}1+x, & if\;x\leq 2\\5-x, & if\;x>2\end{array}\right.$at $x=2$.
LHD at $x=2$
$\qquad\qquad=\lim\limits_{\large x\to 2^-}\large\frac{f(x)-f(2)}{x-2}$
$\qquad\qquad=\lim\limits_{\large x\to 2^-}\large\frac{(1+x)-(1+2)}{x-2}$
$\qquad\qquad=\lim\limits_{\large x\to 2^-}\large\frac{x-2}{x-2}$
Step 2:
LHD at $x=2$
$\qquad\qquad=\lim\limits_{\large x\to 2^+}\large\frac{f(x)-f(2)}{x-2}$
$\qquad\qquad=\lim\limits_{\large x\to 2^+}\large\frac{(5-x)-(5-2)}{x-2}$
$\qquad\qquad=\lim\limits_{\large x\to 2^+}\large\frac{2-x}{x-2}$
Hence LHD at $x=2 \neq$ RHD at $x=2$.
So $f(x)$ is not differentiable at $x=2$
answered Jun 27, 2013 by sreemathi.v

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, AIPMT Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App