- #1
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
$x$ and $y$ are two real numbers such that $x^3-y^3=2$ and $x^5-y^5\ge 4$.
Prove that $x^2+y^2\gt 2$.
Prove that $x^2+y^2\gt 2$.
anemone said:$x$ and $y$ are two real numbers such that $x^3-y^3=2$ and $x^5-y^5\ge 4$.
Prove that $x^2+y^2\gt 2$.
anemone said:$x$ and $y$ are two real numbers such that $x^3-y^3=2$ and $x^5-y^5\ge 4$.
Prove that $x^2+y^2\gt 2$.
The inequality between x and y is x>y.
By rearranging the first equation, we get x^3=2+y^3. Substituting this into the second equation, we get (2+y^3)^5-y^5≥ 4. Expanding and simplifying this inequality, we get 32+80y^3+80y^6+40y^9+10y^12+y^15-y^5≥ 4. Rearranging and simplifying further, we get y^15+10y^12+40y^9+80y^6+80y^3-28≥ 0. This inequality can be proven true for all real values of y, meaning that x>y.
The given equations provide a relationship between x and y that can be used to manipulate and prove the inequality. By substituting one equation into the other, we can simplify and rearrange to ultimately show that x>y.
No, there are no restrictions on the values of x and y. The inequality holds true for all real values of x and y.
The inequality x>y can be represented as a shaded region above the line y=x on a coordinate plane. This is because all points on this region have a greater x-value than y-value. The given equations can also be graphed and the intersection point can be used to show the inequality.