구분구적 조건

$f \left( x \right) = ax \left( x - 3 \right)$으로 잡으면 문제에서 $\displaystyle \frac{k}{n} \rightarrow x$ , $\displaystyle \frac{1}{n} \rightarrow dx$ , $\displaystyle \lim\limits _{n \rightarrow \infty} {\sum\limits _{k = 1} ^{n}} \rightarrow \int _{0} ^{1} {}$로 변형 $\displaystyle \lim\limits _{n \rightarrow \infty} {\frac{1}{n} \sum\limits _{k = 1} ^{n} f \left( \frac{k}{n} \right)} = \frac{7}{6}$ 에서 $\displaystyle \rightarrow \int _{0} ^{1} {f \left( x \right) dx} = a \int _{0} ^{1} {x ^{2} - 3 x} dx$$\displaystyle = a \left[ \frac{x ^{3}}{3} - \frac{3}{2} x ^{2} \right] _{0} ^{1} = - \frac{7}{6} a = \frac{7}{6}$ ∴ $a = - 1$ $f \left( x \right) = - x ^{2} + 3 x$ 를 $x$관하여 미분하면 $f' \left( x \right) = - 2 x + 3$ ∴ $f' \left( 0 \right) = 3$