α(x−1)<xα−1<αxα−1(x−1)\alpha(x-1) < x^{\alpha} - 1 < \alpha x^{\alpha-1}(x-1)α(x−1)<xα−1<αxα−1(x−1) f(b)−f(a)b−a=f(c)′\frac{f_{(b)} - f_{(a)}}{b-a} = f'_{(c)}b−af(b)−f(a)=f(c)′ から f(x)=xαb=x、 a=0f_{(x)} = x^{\alpha} \qquad b = x、\ a = 0f(x)=xαb=x、 a=0 f(x)′=αxα−1f'_{(x)} = \alpha x^{\alpha - 1}f(x)′=αxα−1 区間 [1,x][1, x][1,x] から平均値の定理より xα−1αx−1=αCα−1①\frac{x^{\alpha} - 1^{\alpha}}{x - 1} = \alpha C^{\alpha - 1} \qquad ①x−1xα−1α=αCα−1① C の範囲は 1<C<x1 < C < x1<C<x なので 1α−1<Cα−1<xα−1α<αCα−1<αxα−11^{\alpha - 1} < C^{\alpha - 1} < x^{\alpha - 1} \qquad \alpha < \alpha C^{\alpha - 1} < \alpha x^{\alpha - 1}1α−1<Cα−1<xα−1α<αCα−1<αxα−1 これを①に代入して α<xα−1x−1<αxα−1∴α(x−1)<xα−1<αxα−1(x−1)\alpha < \frac{x^{\alpha} - 1}{x - 1} < \alpha x^{\alpha - 1} \qquad \therefore \alpha(x-1) < x^{\alpha} - 1 < \alpha x^{\alpha-1}(x-1)α<x−1xα−1<αxα−1∴α(x−1)<xα−1<αxα−1(x−1)