A function $f$ is defined by $f ( x ) = e ^ { x }$ if $x < 1$ and $f ( x ) = \log _ { e } ( x ) + a x ^ { 2 } + b x$ if $x \geq 1$. Here $a$ and $b$ are unknown real numbers. Can $f$ be differentiable at $x = 1$ ? (A) $f$ is not differentiable at $x = 1$ for any $a$ and $b$. (B) There exist unique numbers $a$ and $b$ for which $f$ is differentiable at $x = 1$. (C) $f$ is differentiable at $x = 1$ whenever $a + b = e$. (D) $f$ is differentiable at $x = 1$ regardless of the values of $a$ and $b$.
A function $f$ is defined by $f ( x ) = e ^ { x }$ if $x < 1$ and $f ( x ) = \log _ { e } ( x ) + a x ^ { 2 } + b x$ if $x \geq 1$. Here $a$ and $b$ are unknown real numbers. Can $f$ be differentiable at $x = 1$ ?\\
(A) $f$ is not differentiable at $x = 1$ for any $a$ and $b$.\\
(B) There exist unique numbers $a$ and $b$ for which $f$ is differentiable at $x = 1$.\\
(C) $f$ is differentiable at $x = 1$ whenever $a + b = e$.\\
(D) $f$ is differentiable at $x = 1$ regardless of the values of $a$ and $b$.