Challenge Level

This question is about the family of relations given by $x^n+y^n=1$

(1) Explain the features of the graph of the relation $|x|+|y|=1$.

(2) Prove that $${n\over n+1} \leq {1\over 2^{1/n}} < 1 $$

(3) Consider the family of relations $x^n+y^n=1$ in the first quadrant.

Choose one particular value of $n$ and show that $y$ decreases as $x$ increases.

Show that, for each value of $n$, the graph lies entirely outside the square bounded by the lines $$x=0, \ x={n\over n+1},\ y=0,\ y={n\over n+1}$$ and inside the square bounded by the lines $$x=0,\ x=1,\ y=0,\ y=1.$$

(4) Sketch some graphs in all four quadrants of the family of relations $|x|^n+|y|^n=1$ for even values of $n$ and explain why the graphs get closer to a square shape as $n\to \infty$.

(5) Plot the graph of $x^3+y^3=1$ in all four quadrants. Why do the graphs of the relations $x^n+y^n=1$ differ according to whether $n$ is odd or even?