치환적분과 함숫값

[출제의도] 치환적분을 이용하여 조건을 만족시키는 함수의 함숫값을 구할 수 있는가? 조건 (가)에서 $\displaystyle \int _{} ^{} {2 \left\{ f ( x ) \right\} ^{2} f' ( x )} dx$ $\displaystyle = \int _{} ^{} {\left\{ f ( 2 x + 1 ) \right\} ^{2} f' ( 2 x + 1 ) dx}$ 이므로 $\displaystyle \frac{2}{3} \left\{ f ( x ) \right\} ^{3} = \frac{1}{3} \left\{ f ( 2 x + 1 ) \right\} ^{3} \times \frac{1}{2} + C$ (단, $C$는 적분상수) $\left\{ f ( 2 x + 1 ) \right\} ^{3} = 4 \left\{ f ( x ) \right\} ^{3} + C' ( C' = - 6 C )$ $\cdots \text{㉠}$ ㉠에 $x = - 1$을 대입하면 $\left\{ f ( - 1 ) \right\} ^{3} = 4 \left\{ f ( - 1 ) \right\} ^{3} + C'$ 에서 $C' = - 3 \left\{ f ( - 1 ) \right\} ^{3} \cdots \text{㉡}$ ㉠에 $\displaystyle x = - \frac{1}{8}$을 대입하면 $\displaystyle \left\{ f \left( \frac{3}{4} \right) \right\} ^{3} = 4 \left\{ f \left( - \frac{1}{8} \right) \right\} ^{3} + C' = 4 + C'$ ㉠에 $\displaystyle x = \frac{3}{4}$을 대입하면 $\displaystyle \left\{ f \left( \frac{5}{2} \right) \right\} ^{3} = 4 \left\{ f \left( \frac{3}{4} \right) \right\} ^{3} + C' = 16 + 5 C'$ ㉠에 $\displaystyle x = \frac{5}{2}$을 대입하면 $\displaystyle \left\{ f \left( 6 \right) \right\} ^{3} = 4 \left\{ f \left( \frac{5}{2} \right) \right\} ^{3} + C'$ $2 ^{3} = 64 + 21 C'$ $\displaystyle C' = - \frac{8}{3} \cdots \text{㉢}$ ㉡, ㉢에서 $\displaystyle - 3 \left\{ f ( - 1 ) \right\} ^{3} = - \frac{8}{3}$ 이므로 $\displaystyle \left\{ f ( - 1 ) \right\} ^{3} = \frac{8}{9}$ 따라서 $\displaystyle f ( - 1 ) = \frac{2 \sqrt[3]{3}}{3}$