Let #a, b > 0, a+b = 1, n>1# Show that #(a+1/a)^n + (b+1/b)^n >= 5^n/n^(n-1)#?
#a, b > 0, a+b = 1, n>1#
#(a+1/a)^n + (b+1/b)^n >= 5^n/n^(n-1)#
This problem can be stated as a minimization problem.
Analyzing the problem we see due to the symmetry, that
Now substituting those values into the objective function we have
It is necessary to analyze for the minimum of
This matrix has as characteristic polynomial
Note. The point
Another point of view.
Due to the symmetry the minimization problem requires that
Suppose instead that symmetry does not occur
and as can be verified,