Câu hỏi của hello sunshine

Question

Bài 2: Cho x, y, z dương thỏa mãn: x + y + z = 1. CMR:

$\sqrt{2x+1}+\sqrt{2y+1}+\sqrt{2z+1}< 4$

Bài 3: Cho ab + bc + ca = 9. CMR:

a4 + b4 + c4 $\ge27$

Nguyễn Việt Lâm · Accepted Answer

2.

Áp dụng BĐT $\left(a+b+c\right)^2\le3\left(a^2+b^2+c^2\right)$

$\Rightarrow VT=\sqrt{2x+1}+\sqrt{2y+1}+\sqrt{2z+1}\le\sqrt{3\left(2x+1+2y+1+2z+1\right)}$

$\Rightarrow VT\le\sqrt{3\left[2\left(x+y+z\right)+3\right]}=\sqrt{15}< \sqrt{16}=4$ (đpcm)

3.

$VT=a^4+b^4+c^4\ge\frac{1}{3}\left(a^2+b^2+c^2\right)^2\ge\frac{1}{3}\left[3\left(ab+bc+ca\right)\right]^2=27$

Dấu "=" xảy ra khi $a=b=c=\sqrt{3}$Câu trả lời: 2.

Áp dụng BĐT $\left(a+b+c\right)^2\le3\left(a^2+b^2+c^2\right)$

$\Rightarrow VT=\sqrt{2x+1}+\sqrt{2y+1}+\sqrt{2z+1}\le\sqrt{3\left(2x+1+2y+1+2z+1\right)}$

$\Rightarrow VT\le\sqrt{3\left[2\left(x+y+z\right)+3\right]}=\sqrt{15}< \sqrt{16}=4$ (đpcm)

3.

$VT=a^4+b^4+c^4\ge\frac{1}{3}\left(a^2+b^2+c^2\right)^2\ge\frac{1}{3}\left[3\left(ab+bc+ca\right)\right]^2=27$

Dấu "=" xảy ra khi $a=b=c=\sqrt{3}$

Chương I - Căn bậc hai. Căn bậc ba