Question

Let $f\left(x\right)=\{\begin{array}{ccc}x+1,& when,& x<2\\ 2x-1,& when& x\ge 2\end{array},then{f}^{\prime }\left(2\right)=$

• A. 1
• B. 2
• C. Does not exist

Answer 1

The correct option is C Does not exist

$R{f}^{\prime }\left(2\right)={lim}_{h\to 0}\frac{f(2+h)-f\left(2\right)}{h}$
$={lim}_{h\to 0}\frac{2(2+h)-1-(4-1)}{h}={lim}_{h\to 0}\frac{4+2h-1-3}{h}=2$
$andL{f}^{\prime }\left(2\right)={lim}_{h\to 0}\frac{f(2-h)-f\left(2\right)}{-h}={lim}_{h\to 0}\frac{2-h+1-3}{-h}=1$
Since Right hand derivative and Left hand derivative are not same, f'(2) does not exist.