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Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

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How Many Solutions?

Find all the solutions to the this equation.

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After Thought

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

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