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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?

How Many Solutions?

Find all the solutions to the this equation.

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