Question

Chứng minh bất đẳng thức x⁴+16>=2x³+8x

Answer 1

Giả sử : \(x^4+16\ge2x^3+8x\)

\(\Leftrightarrow x^4-2x^3-8x+16\ge0\)

\(\Leftrightarrow\left(x^4-2x^3\right)-\left(8x-16\right)\ge0\)

\(\Leftrightarrow x^3\left(x-2\right)-8\left(x-2\right)\ge0\)

\(\Leftrightarrow\left(x-2\right)\left(x^3-8\right)\ge0\)

\(\Leftrightarrow\left(x-2\right)\left(x-2\right)\left(x^2+2x+4\right)\ge0\)

\(\Leftrightarrow\left(x-2\right)^2\left(x^2+2x+4\right)\ge0\) ( luôn đúng )

⇒ đpcm