Question

Let $f\left(x\right)=\{\begin{array}{c}{e}^x,x<1\\ {\log }_{e}x+a{x}^2+b,x\ge 1\end{array}$

where $a,b\in R.$ If $f$ is differentiable at $x=1$, then

• A. for unique values of $a$ and $b$
• B. for any values of $a$ and $b$
• C. whenever $a+b=e$
• D. for no values of $a$ and $b$

Answer 1

The correct option is A for unique values of $a$ and $b$

If $f$ is differentiable at $x=1$, then $f$ must be continuous at $x=1$

LHL at $x=1:$
$\underset{x\to 1^{-}}{lim}f\left(x\right)=e$
RHL at $x=1:$
$\underset{x\to 1^{+}}{lim}f\left(x\right)=a+b$
$f\left(1\right)=a+b$

$\Rightarrow a+b=e\cdots \left(1\right)$

LHD at $x=1:$
$f^{\prime }\left(1^{-}\right)=e$
RHD at $x=1:$
$f^{\prime }\left(1^{+}\right)=2a+1$

If $f$ is differentiable at $x=1$,
$2a+1=e$
$\Rightarrow a=\frac{e-1}{2}$
From $\left(1\right),$ we get
$b=\frac{e+1}{2}$
$\therefore f$ is differentiable for unique values of $a$ and $b$