微分運用概念？？恆值定理..請協助

Question

3 + 1/28 ≦ 3√28 ≦ 3 + 1/27 以恆值定律證明。(3√28 是開3次根號)

Answer 1

設f(x)=x1/3
在[27, 28]內f(x)連續又可微分
f'(x) = 1/3 x-2/3 = 1/(3 x2/3)
=>可用平均值定理(均值定理)
=> [f(28)-f(27)]/(28-27) = f'(c), c介於27,28之間,
即 281/3 - 271/3 = 1/(3 c2/3) ---(A)
又 27 1/(3*282/3) < 1/(3 c2/3) < 1/(3* 272/3)
而1/(3* 282/3)> 1/28, 1/(3* 272/3)=1/27
代入(A)式, 則281/3=3 + 1/(3*c2/3)
故 3 + 1/28 < 281/3 < 3 + 1/27