# 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?

##### 3 Answers

There is no such point, as far as my math goes.

#### Explanation:

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.

#### Explanation:

We see that

In the first case,

In the second case,

but

Concluding, there is not such a tangent.

The answer from Dr, Cawa K, x = 1/e, is precise.

#### Explanation:

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. .