Expand and simplify (x+y)2(x + y)^2(x+y)2.
Sarah claims that for any non-zero real number kkk, the sum of kkk and its reciprocal, k+1kk + \frac{1}{k}k+k1, is always greater than or equal to 2.
Sarah tests two values: When k=3k = 3k=3, 3+13=3.3˙≥23 + \frac{1}{3} = 3.\dot{3} \ge 23+31=3.3˙≥2. When k=1k = 1k=1, 1+11=2≥21 + \frac{1}{1} = 2 \ge 21+11=2≥2.
Provide a counter-example to show that Sarah's claim is incorrect.
Given that xxx and yyy are distinct positive real numbers, use proof by contradiction to prove that:
xy+yx>2 \frac{x}{y} + \frac{y}{x} > 2 yx+xy>2
