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Folium of Descartes

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.

Witch of Agnesi

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

Discrete Trends

Find the maximum value of n to the power 1/n and prove that it is a maximum.

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 $y=x^{1/x}$.