SOLVED Is f(x) differentiable at x=1

Jan 2010
52
2
Given
\(\displaystyle f(x)=\left\{\begin{array}{cc}x^2,&\mbox{ if }
x\leq 1\\\frac{x+1}{2}, & \mbox{ if } x>1\end{array}\right\)

Is f(x) differentiable at x=1?

I know that we have to prove
\(\displaystyle \boxed{ \lim_{\Delta x \to 0}\frac{f(1+\Delta x)-f(1)}{\Delta x}}\)

exist/does not exist at x=1. But how do I begin? It's a piece-wise function.

Thanks for your help.
 

Jester

MHF Helper
Dec 2008
2,470
1,255
Conway AR
What I might suggest is using the alternate definition

\(\displaystyle \lim \limits_{x \to a} \dfrac{f(x)-f(a)}{x - a}\).

Then as \(\displaystyle x \to 1^{-}\) and \(\displaystyle x \to 1^{+} \) you can use the appropriate branch.