Question

C/m  bất đẳng thức sau

\(\left(a+b\right)\left(a^3+b^3\right)\le2\left(a^4+b^4\right)\)

Answer 1

\(\left(a+b\right)\left(a^3+b^3\right)\le2\left(a^4+b^4\right)\)

\(\Leftrightarrow a^4+a^3b+ab^3+b^4\le2a^4+2b^4\)

\(\Leftrightarrow a^3b-a^4+ab^3-b^4\le0\)

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

\(\Leftrightarrow\left(a-b\right)\left(b^3-a^3\right)\le0\)

\(\Leftrightarrow-\left(a-b\right)^2\left(a^2+ab+b^2\right)\le0\) (luôn đúng)

Vậy...

Answer 2

a + b a + b ≤ 2 a + b

⇔a + a b + ab + b ≤ 2a + 2b

⇔a b − a + ab − b ≤ 0

⇔a b − a + b a − b ≤ 0

⇔ a − b b − a ≤ 0

⇔− a − b a + ab + b ≤ 0 

tự kết luận