Expand and simplify (5x−2y)2(5x - 2y)^2(5x−2y)2.
An environmental scientist claims that for any non-zero real number vvv, the sum of vvv and its scaled reciprocal, v+49vv + \frac{49}{v}v+v49, is always greater than or equal to 14.
The scientist tests two values: When v=7v = 7v=7, 7+497=14≥147 + \frac{49}{7} = 14 \ge 147+749=14≥14. When v=10v = 10v=10, 10+4910=14.9≥1410 + \frac{49}{10} = 14.9 \ge 1410+1049=14.9≥14.
Provide a counter-example to show that the scientist's claim is incorrect.
Given that xxx and yyy are positive real numbers such that 5x≠2y5x \neq 2y5x=2y, use proof by contradiction to prove that:
5x2y+2y5x>2 \frac{5x}{2y} + \frac{2y}{5x} > 2 2y5x+5x2y>2
