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A function $f(x)$ is defined as

$f(x)= \begin{cases}  e^{x}, & x < 1  \\  \text{In } x+ax^{2}+bx, & x\geq 1 \end{cases}$, where $x \in \mathbb{R}$

Which one of the following statement is TRUE?

  1. $f(x)$ is NOT differentiable at $x=1$ for any values of $a$ and $b$.
  2. $f(x)$ is differentiable at $x=1$ for the unique values of $a$ and $b$.
  3. $f(x)$ is differentiable at $x=1$ for all values of $a$ and $b$ such that $a+b=e$.
  4. $f(x)$ is differentiable at $x=1$ for all values of $a$ and $b$.
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