Mathematical Induction ~~ 超難

∛3 > n^(1/n) , the required is that n ≥ 4When n = 4 , ∛3 > ∜4 is true ,Assuming that ∛3 > k^(1/k) , the required is that k ≥ 4 ,
i.e. 3^k > k³ When n = k+1 ,3^(k+1) = 3(3^k)
> 3k³
= k³ + k(k²) + k²(k)
= k³ + k(k²) + (k²-1)(k) + k
> k³ + 3(k²) + 3(k) + 1 = (k+1)³ for k ≥ 4The inequality is true for n = k+1.

Therefore , by the principle of mathematical induction ,
the inequality is true for n ≥ 4.i.e. 3^n > n³ ∛3 > n^(1/n) for n > 3.

YipKa Lok既方法係岩, 不過唔好意思, 我只能夠將第一位答中選出最佳解答.

Let y = x^(1/x)
=> dy/dx = (x^(1/x))[1-ln x]/(x^2)
=> [ >0 when x
=[ =0 when x=e ]
=> [ <0 when x>e ] (since x^(1/x) , x^2 > 0 for all x>0)
i.e. x^(1/x) is max. at x=e and decreasing when x > e

2011-06-24 12:06:24 補充：
hence, e^(1/e) > 3^(1/3) > n^(1/n) for all positive integer n>3.