Exponential Trend

Age 16 to 18

Challenge Level Yellow star

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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