Question

chứng minh bất đẳng thức 2a^3+8a<=a^4+16

Answer 1

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

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

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

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

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

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 !!!

Answer 2

cảm ơn bạn