Question

If $f\left(x\right)=\{\begin{array}{cc}{e}^x,& x<2\\ a+bx,& x\ge 2\end{array}$ is differentiable for all $x\in R,$ then which of the following is/are correct

• A. $b=e^2$
• B. $a+2b=e^2$
• C. $a+b=0$
• D. $a+b=e$

Answer 1

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

$a+b=0$

Since $f\left(x\right)$ is differentiable for all $x\in R.$
It has to be differentiable at $x=2.$
From given function we have
$f^{\prime }\left(x\right)=\{\begin{array}{cc}{e}^x,& x<2\\ b,& x>2\end{array}$
Now,
$R.H.D.$ at $x=2\Rightarrow f^{\prime }\left(2^{+}\right)=b$
$L.H.D.$ at $x=2\Rightarrow f^{\prime }\left(2^{-}\right)=e^2$
$\Rightarrow b=e^2\cdots \left(i\right)$

Also, for continuity of $f\left(x\right)$ at $x=2$, we have $f\left(2^{-}\right)=f\left(2^{+}\right)=f\left(2\right)$
$\Rightarrow e^2=a+2b\cdots \left(ii\right)$
Now from $\left(i\right)$ and $\left(ii\right)$, we have
$a=-e^2$