On the coordinate plane, for a cubic function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x$ and a real number $t$, let P be the point where the tangent line to the curve $y = f ( x )$ at the point $( t , f ( t ) )$ intersects the $y$-axis. Let $g ( t )$ be the distance from the origin to point P. The function $f ( x )$ and the function $g ( t )$ satisfy the following conditions. (가) $f ( 1 ) = 2$ (나) The function $g ( t )$ is differentiable on the entire set of real numbers. What is the value of $f ( 3 )$? (Here, $a , b$ are constants.) [4 points] (1) 21 (2) 24 (3) 27 (4) 30 (5) 33
On the coordinate plane, for a cubic function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x$ and a real number $t$, let P be the point where the tangent line to the curve $y = f ( x )$ at the point $( t , f ( t ) )$ intersects the $y$-axis. Let $g ( t )$ be the distance from the origin to point P. The function $f ( x )$ and the function $g ( t )$ satisfy the following conditions.\\
(가) $f ( 1 ) = 2$\\
(나) The function $g ( t )$ is differentiable on the entire set of real numbers.\\
What is the value of $f ( 3 )$? (Here, $a , b$ are constants.) [4 points]\\
(1) 21\\
(2) 24\\
(3) 27\\
(4) 30\\
(5) 33