# Let #a, b > 0, a+b = 1, n>1# Show that #(a+1/a)^n + (b+1/b)^n >= 5^n/n^(n-1)#?

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Let #a, b > 0, a+b = 1, n>1#

Show that #(a+1/a)^n + (b+1/b)^n >= 5^n/n^(n-1)#

Let

Show that

##### 2 Answers

#### Answer:

See below.

#### Explanation:

This problem can be stated as a minimization problem.

Calling

Find

subjected to

If

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

The Hessian

at

This matrix has as characteristic polynomial

Note. The point

we have

#### Answer:

See below.

#### Explanation:

Another point of view.

Making now

Due to the symmetry the minimization problem requires that

or

and

Suppose instead that symmetry does not occur

then

and

Now

and as can be verified,