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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
$$\lim_{x\to \infty} x^{1/x} = \lim_{n\to \infty} n^{1/n}.$$
Exponential Trend
Age 16 to 18
Challenge Level 
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 $y=x^{1/x}$.
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