Prove that for any two real numbers x , y with x ≠ 0 2 y ≤ y 2 x 2 + x 2

Hint:Explore the following expression ( x − y x ) 2

By AM-GM we obtain: y 2 x 2 + x 2 ≥ 2 y 2 x 2 ⋅ x 2 = 2 | y | ≥ 2 y .Done!

Hint:Explore the following expression ( x − y x ) 2

By AM-GM we obtain: y 2 x 2 + x 2 ≥ 2 y 2 x 2 ⋅ x 2 = 2 | y | ≥ 2 y .Done!

Best solutions to - "Prove that for any two real numbers x..."

Hana Buck

2022-09-23

Prove that for any two real numbers

x, y

with

x \neq 0

2 y \leq \frac{y^{2}}{x^{2}} + x^{2}

Kaiden Stevens

Beginner2022-09-24Added 12 answers

Hint:
Explore the following expression

{(x - \frac{y}{x})}^{2}

mundocromadomg

Beginner2022-09-25Added 3 answers

By AM-GM we obtain:

\frac{y^{2}}{x^{2}} + x^{2} \geq 2 \sqrt{\frac{y^{2}}{x^{2}} \cdot x^{2}} = 2 | y | \geq 2 y .

Done!

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