Is there any point #(x, y)# on the curve #y=x^(x(1+1/y)), x > 0,# at which the tangent is parallel to the x-axis?
There is no such point, as far as my math goes.
First, let's consider the conditions of the tangent if it is parallel to the
Therefore, we must first start by finding the derivative of this monstrous equation, which can be accomplished through implicit differentiation:
Using the sum rule, chain rule, product rule, quotient rule, and algebra, we have:
Wow...that was intense. Now we set the derivative equal to
Interesting. Now let's plug in
Since this is a contradiction, we conclude that there are no points meeting this condition.
There not exists such a tangent.
We see that
In the first case,
In the second case,
Concluding, there is not such a tangent.
The answer from Dr, Cawa K, x = 1/e, is precise.
I had proposed this question to get this value precisely. Thanks to
Dr, Cawas for a decisive answer that approves the revelation that
the double precision y' remains 0 around this interval. y is
continuous and differentiable at x = 1/e. As both the 17-sd double
precision y and y' are 0, in this interval around x = 1/e, it was a
conjecture that x-axis touches the graph in between. And now, it is
proved. I think that the touch is transcendental. .