극한 조건 삼차함수

$\displaystyle \lim\limits _{x \rightarrow 2} \frac{f ( x )}{( x - 2 ) ^{2}} = 3$에서 $\displaystyle \lim\limits _{x \rightarrow 2} ( x - 2 ) ^{2} = 0$이므로 $\displaystyle \lim\limits _{x \rightarrow 2} f ( x ) = 0$ ∴ $f ( x ) = ( x - 2 ) ^{2} ( ax + b )$ (단, a, b 는 상수) ∴$\displaystyle \lim\limits _{x \rightarrow 2} \frac{f ( x )}{( x - 2 ) ^{2}} = \lim\limits _{x \rightarrow 2} \frac{( x - 2 ) ^{2} ( ax + b )}{( x - 2 ) ^{2}}$ $\displaystyle = \lim\limits _{x \rightarrow 2} ( ax + b )$ $= 2 a + b = 3$ ……………………… ㉠ $f ( 3 ) = 5$에서 $f ( 3 ) = ( 3 - 2 ) ^{2} ( 3 a + b ) = 5$ ∴ $3 a + b = 5$ ……………………… ㉡ ㉠, ㉡을 연립하여 풀면 $a = 2$, $b = - 1$ 따라서, $f ( x ) = ( x - 2 ) ^{2} ( 2 x - 1 )$이므로 $f' ( x ) = 2 ( x - 2 ) ( 2 x - 1 ) + ( x - 2 ) ^{2} \cdot 2$ $= 2 ( x - 2 ) ( 3 x - 3 )$ ∴ $f' ( 3 ) = 2 ( 3 - 2 ) ( 3 \cdot 3 - 3 )$=12