MHB Inequality of positive real numbers

[anemone]

If $x$ and $y$ are positive real numbers, prove that $4x^4+4y^3+5x^2+y+1\ge 12xy$.

 

[lfdahl]

Spoiler: Suggested solution

Applying the arithmetic-mean geometric-mean inequality three* times:

From $4x^4 + 1 \geq^* 2 \sqrt{4x^4} = 4x^2$ and $4y^3+y \geq^* 2\sqrt{4y^4}= 4y^2$ - we have

$4x^4+4y^3+5x^2+y+1 \geq 9x^2 + 4y^2 \geq^* 2\sqrt{36x^2y^2} = 12xy$.