Ta có: \(2x^2+2x+1\)

\(=2\left(x^2+x+\frac{1}{2}\right)\)

\(=2\left(x^2+2\cdot x\cdot\frac{1}{2}+\frac{1}{4}+\frac{1}{4}\right)\)

\(=2\left(x+\frac{1}{2}\right)^2+\frac{1}{2}\)

Ta có: \(\left(x+\frac{1}{2}\right)^2\ge0\forall x\)

\(\Rightarrow2\left(x+\frac{1}{2}\right)^2\ge0\forall x\)

\(\Rightarrow2\left(x+\frac{1}{2}\right)^2+\frac{1}{2}\ge\frac{1}{2}\forall x\)

hay \(2x^2+2x+1>0\forall x\)(đpcm)

\(=x^2+6x+5+x^3-8-x^3-x^2+2x\)

=8x-3

\(x^2\ge2x-1\Leftrightarrow x^2-2x+1\ge0\Leftrightarrow\left(x-1\right)^2\ge0\)

Có: \(\left(x-1\right)^2\ge0,\forall x\)

\(\rightarrow x^2-2x+1\ge0,\forall x\)

\(\Rightarrow x^2\ge2x-1,\forall x\)

Nhớ tick mik nha

\(2a^3+8a\le a^4+16\)

\(\Leftrightarrow2a^3+8a-a^4-16\le0\)

\(\Leftrightarrow\left(2a^3-a^4\right)+\left(8a-16\right)\le0\)

\(\Leftrightarrow-a^3\left(a-2\right)+8\left(a-2\right)\le0\)

\(\Leftrightarrow-\left(a-2\right)\left(a^3-8\right)\le0\Leftrightarrow-\left(a-2\right)^2\left(a^2+2a+4\right)\le0\)

TA THẤY : \(\left(a-2\right)^2\left(a^2+2a+4\right)\ge0\)\(\Leftrightarrow-\left(a-2\right)^2\left(a^2+2a+4\right)\le0\)\(\Leftrightarrow2a^3+8a\le a^4+16\left(dpcm\right)\)

DẤU " = " XẢY RA KHI X = 2

TK CHO MK NKA !!!

cảm ơn bạn