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

(2) Prove that

${\displaystyle \frac{n}{n+1}\le \frac{1}{2^{1/n}}<1}$

(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

${\displaystyle x=0,\hspace{1em}x=\frac{n}{n+1},\hspace{1em}y=0,\hspace{1em}y=\frac{n}{n+1}}$

and inside the square bounded by the lines

${\displaystyle x=0,\hspace{1em}x=1,\hspace{1em}y=0,\hspace{1em}y=1.}$

(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 tho whether $n$ is odd or even?