Matematické Fórum

Skrytý text:

Any inference from graphs should not be taken as a proof in mathematical analysis.

My suggestion:
Taking into consideration only x from (0, \infty) (because of the logarithm), I would find the tangent of the logarithm in [1,0] and the tangent of the exponential in [0,1]. Then two properties are useful:

1) These tangents are parallel, so they have no intersection.

2) Logarithm here is concave whereas the exponential is convex. From the property of concave (or convex) functions and their tangents, we can deduce the inexistence of solutions of the original equation.

Specific theorems for point 2) should be added.

Edit:

To be more specific, let's define



Then

,

where x-1 and x+1 are the tangents. The first and last inequalities are the properties mentioned in 2), the middle inequality is obvious.

Skrytý text:


My description was a bit vague, so to be more specific:

For a strictly concave function we know that all the points lie "below" (or exactly on) any given tangent. The theorem could be stated as follows:

Let G be a convex set and . Then is strictly concave if and only if



Where the right hand side is the value of the point on the tangent for .