例1已知f(n)=n~(n+1),g(x)=(n+1)~n,n∈N*.求证:当n≥3,n∈N*时,f(n)>g(x).本题用数学归纳法可以证明.但是用加强命题,再利用导数方法解决则是另外一种风味.证明对于上述命题,我们可以先加强命题x≥3,x∈R时,有x~(x+1)>(x+1)~x.即(x+1)lnx>xln(x+1),因为x≥3,lnx>0,ln(x+1)>0, Example 1 Known f (n) = n ~ (n + 1), g (x) = (n + 1) ~ n, n∈N * Proof: When n≥3, n∈N *, f This problem can be proved by mathematical induction, but it is another flavor to use enhanced proposition and derivative method to prove that for the above proposition, we can strengthen the proposition x≥3, x∈R (X + 1) lnx> xln (x + 1) since x ≧ 3, lnx> 0, ln (x + 1)> 0,