is there any horizantal tangent line in this function?

Question

y=x^sqrtx

Answer 1

Assume x > 0

 

y = x^√(x)

 

Take the log of both sides: ln(y) = √(x) * ln(x)

 

Do implicit differentiation: y'/y = (1/(2√(x)) * ln(x) + 1/√(x)

 

                                                = (1/√(x)) * (ln(x)/2 + 1)

 

A horizontal tangent occurs when y' = 0.

Assume x > 0 because of the square root and 1/x.

 

When y' = 0, (1/√(x)) * (ln(x)/2 + 1) = 0

 

                     ln(x)/2 + 1 = 0

 

                     ln(x)/2 = -1

 

                      ln(x) = -2

 

                       x = e^-2

 

Now y = x^√(x) = (e^-2)^{e^-1} = e^{-2e^-1}

 

∴ The point (x, y) = (e^-2 , e^{-2e^-1}) ≈ (0.1353, 0.4791) has a horizontal tangent.

 

Answer 2

yes it have horizontal tangent line at x=e^-2