x^n+y^n=1 even powers
This question is about the family of relations given by xn+yn=1

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

(2) Prove that
n
n+1
 £ 1
21/n
 < 1

(3) Consider the family of relations xn+yn=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
n+1
 , y=0, y= n
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® ¥.

(5) Plot the graph of x3+y3=1 in all four quadrants. Why do the graphs of the relations xn+yn=1 differ according tho whether n is odd or even?