Question

In Exercises \(51-54,\) find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. (Hint: Some of the limits required in finding asymptotes have been found in previous exercises.) \(y=x^{1 / x}, \quad x>0\)

Short answer

The function \(y = x^{1/x}\) has a relative minimum at \(x = e\) and no maximum. It has a horizontal asymptote at \(y = 1\) and does not have a vertical asymptote.

Step-by-step solution

Find the derivative of the function

The first step involves finding the derivative of the function. Unfortunately, this function is not straightforward. Using the function transformation \(y = e^{(1/x)*ln(x)}\) may simplify the process. Then, differentiate the function using chain rule. The derivative of \(y\) with respect to \(x\) is \(dy/dx = -\frac{ln(x)}{x^2}e^{(1/x) * ln(x)} + \frac{1}{x^2} e^{(1/x)*ln(x)} = x^{1/x}(\frac{1}{x^2} -\frac{ln(x)}{x^2})\).

Find the critical points

By setting the derivative of the function equal to 0 and solving for \(x\), the critical points of the function are found. These are the points where the function reaches a local maximum or minimum. By solving \(dy/dx = 0\), we find one critical point, \(x = e\).

Find the second derivative

The second derivative allows us to identify whether the critical point is a local minimum or a maximum. Applying the same transformation and differentiating again gives \(d^2y/dx^2 = -x^{1/x}*\frac{2-ln(x)}{x^3}\).

Applying the second derivative test

Evaluate the second derivative at the critical point \(x=e\). Which is positive, indicating this point as a relative minimum.

Find the asymptotes

The horizontal asymptote can be found by calculating the limit as \(x\) approaches infinity. Keeping in mind that \(e^{ln(1/x)} = 1/x\), the limit of the function as \(x -> + \infty\) is \(1\). For the vertical asymptote, check for the limit as \(x\) approaches 0 from the right because the function is given for \(x > 0\). That gives an undefined value. Hence, there are no vertical asymptotes for this function.

Graph the function using a graphing utility

Finally, the function \(x^{1/x}\) is graphed using a graphing utility. It will confirm the relative minimum and asymptote.