Question

Cho 2x + 3y = 5. CMR: 2x²+ 3y² >= 5 

Answer 1

Ta có:` a^2 + b^2 >= 2ab`.

`-> 2x^2 +2 >= 4x` và `3y^2 + 3 >= 6y`.

`-> 2x^2 + 3y^2 >= 2(2x+3y) - 5`.

`-> 2x^2 + 3y^2 >= 2 . 5 - 5 = 10 - 5 = 5`.

Dấu bằng xảy ra `<=> x = y = 1.`

Answer 2

\(2x+3y=5\Rightarrow y=\dfrac{5-2x}{3}\)

\(\Rightarrow2x^2+3y^2=2x^2+3\left(\dfrac{5-2x}{3}\right)^2=\dfrac{10x^2-20x+25}{3}=\dfrac{10\left(x-1\right)^2}{3}+5\ge5\)

Dấu "=" xảy ra khi \(x=y=1\)