Question

If $f\left(x\right)=\{\begin{array}{c}{e}^x,x<2\\ a+bx,x\ge 2\end{array}$, is differentiable for all $x\in R$ then

• A. $a+b=0$
• B. $a+2b=e^2$
• C. $b=e^2$
• D. none of these

Answer 1

Answer: (A), (B), (C)

$a+2b=e^2$
$b=e^2$
$a+b=0$

For given function to be differentiable, it has to be continuous.
$\Rightarrow f(2-)=f(2+)=f\left(2\right)=>a+2b=e^2$
Also $f^{\prime }(2+)=f^{\prime }(2-)\Rightarrow e^2=b\Rightarrow a=-e^2$