Show that the turning points of

e^{f (x)}

occur for the same values of

x

as the turning points of

f (x)

Find all the turning points of $x^{1 / x}$ for $x > 0$ and decide whether each is a maximum or minimum. Give a sketch of the graph of $y = x^{1 / x}$ for $x > 0$ . Deduce from your sketch that

$lim_{x \to \infty} x^{1 / x} = lim_{n \to \infty} n^{1 / n} .$

Now use the result from the problem Discrete Trends to find this limit.

Show that

$lim_{x \to 0} x^{1 / x} = 0$

by substituting $t = 1 / x$ . Hence find the largest value of $c$ such that the line $y = c$ lies under the graph of $y = x^{1 / x}$ .