평행이동한 삼차함수 적분

[출제의도] 정적분을 활용하여 문제를 해결한다. $g ( 0 ) = 0$이므로 $\displaystyle \lim\limits _{x \rightarrow 0 -} {\frac{g ( x ) - g ( 0 )}{x - 0}}$$\displaystyle = \lim\limits _{x \rightarrow 0 -} {\frac{f \left( x - p \right) - f \left( - p \right)}{x}}$$= f' \left( - p \right)$ $\displaystyle \lim\limits _{x \rightarrow 0 +} {\frac{g ( x ) - g ( 0 )}{x - 0}}$$\displaystyle = \lim\limits _{x \rightarrow 0 +} {\frac{f \left( x + p \right) - f ( p )}{x}}$$= f' \left( p \right)$ $g' ( 0 ) = 0$이므로 $f' \left( - p \right) = f' ( p )$$= 0$ $f' ( x )$는 이차항의 계수가 $3$인 이차식이므로 $f' ( x )$$= 3 \left( x + p \right) \left( x - p \right)$$= 3 x ^{2} - 3 p ^{2}$ 따라서 $f ( x )$$= x ^{3} - 3 p ^{2} x + C$(단, $C$는 적분상수) $f ( 0 ) = 1$이므로 $f ( x )$$= x ^{3} - 3 p ^{2} x + 1$ $x \geq 0$에서 $g ( x )$$= f \left( x + p \right) - f ( p )$이므로 $\displaystyle \int _{0} ^{p} {g ( x )} dx$$\displaystyle = \int _{0} ^{p} {\left\{ f \left( x + p \right) - f \left( p \right) \right\}} dx$ $\displaystyle = \int _{0} ^{p} {\left( x ^{3} + 3 px ^{2} \right)} dx$$\displaystyle = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} \frac{x ^{4}}{4} + px ^{3} \end{array} \right] _{0} ^{p}$$\displaystyle = \frac{5}{4} p ^{4}$$= 20$ $p > 0$이므로 $p = 2$이고, $f ( x )$$= x ^{3} - 12 x + 1$ 따라서 $f ( 5 )$$= 66$