Dấu "=" xảy ra khi x=y=2; ta có : \(\sqrt[3]{8^x.8^x}=\sqrt[3]{64^x}=4^x\)

\(8^x+8^x+8^2\ge3\sqrt[3]{8^x.8^x.8^2}=12.4^x\)

\(8^y+8^y+8^2\ge12.4^y\)

\(8^z+8^z+8^2\ge12.4^z\)

Cộng 3 vế BĐT trên => đpcm

Một cách khác:

Đặt $(2^x,2^y,2^z)=(a,b,c)\Rightarrow abc=2^{x+y+z}=2^6=64$

Bài toán trở thành:

Cho $a,b,c>0$ sao cho $abc=64$. CMR: $a^3+b^3+c^3\geq 4(a^2+b^2+c^2)$

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Áp dụng BĐT Bunhiacopxky và AM-GM:

$(a^3+b^3+c^3)(a+b+c)\geq (a^2+b^2+c^2)^2$

\(\Rightarrow a^3+b^3+c^3\geq \frac{(a^2+b^2+c^2)^2}{a+b+c}(1)\)

Mà: \(a^2+b^2+c^2\geq \frac{(a+b+c)^2}{3}\geq \frac{(a+b+c).3\sqrt[3]{abc}}{3}=\frac{(a+b+c).3\sqrt[3]{64}}{3}=4(a+b+c)(2)\)

Từ \((1);(2)\Rightarrow a^3+b^3+c^3\geq \frac{(a^2+b^2+c^2).4(a+b+c)}{a+b+c}=4(a^2+b^2+c^2)\) (đpcm)

Vậy.......

ta có:

X⁴ z⁴ y⁴  luôn>0

x-y>=\(\sqrt{2xy}\)  >0

tương tự z-x,  y-z  =>A luôn dương

mũ 4 hay nhan 4 vay

Sửa đề:

Chứng minh rằng:

\(8x+8y+8z\le4^{x+1}+4^{y+1}+4^{y+2}\)

Ta có:

\(8x+8y+8z=8.\left(x+y+z\right)=8.6=48\)(1)

Áp dụng bất đẳng thức AM-GM ta có:

\(4^{x+1}+4^{y+1}+4^{z+1}\ge3\sqrt[3]{4^{x+1}.4^{y+1}.4^{z+1}}\)

\(\Rightarrow4^{x+1}+4^{y+1}+4^{z+1}\ge3\sqrt[3]{4^{x+y+z+3}}\)

\(\Rightarrow4^{x+1}+4^{y+1}+4^{z+1}\ge3\sqrt[3]{4^{6+3}}\)

\(\Rightarrow4^{x+1}+4^{y+1}+4^{z+1}\ge3\sqrt[3]{4^9}\)

\(\Rightarrow4^{x+1}+4^{y+1}+4^{z+1}\ge3.64\)

\(\Rightarrow4^{x+1}+4^{y+1}+4^{z+1}\ge192\)(2)

Dấu "=" sảy ra khi \(x=y=z=2\).

Từ (1) và (2) suy ra:

\(8x+8y+8z\le4^{x+1}+4^{y+1}+4^{y+2}\)(đpcm)

Chúc bạn học tốt!!!

Đề gốc:\(8^x+8^y+8^z\ge4^{x+1}+4^{y+1}+4^{z+1}\)

\(\frac{x^5}{y^4}+\frac{x^5}{y^4}+y+y+y\ge5\sqrt[5]{\frac{x^{10}y^3}{y^8}}=\frac{5x^2}{y}\)

Tương tự: \(\frac{2y^5}{z^4}+3z\ge\frac{5y^2}{z}\) ; \(\frac{2z^5}{x^4}+3x\ge\frac{5z^2}{x}\)

Cộng vế với vế:

\(2\left(\frac{x^5}{y^4}+\frac{y^5}{z^4}+\frac{z^5}{x^4}\right)+3\left(x+y+z\right)\ge5\left(\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\right)\ge5\left(x+y+z\right)\)

\(\Rightarrow2\left(\frac{x^5}{y^4}+\frac{y^5}{z^4}+\frac{z^5}{x^4}\right)\ge2\left(x+y+z\right)\ge2\)

\(\Rightarrow\frac{x^5}{y^4}+\frac{y^5}{z^4}+\frac{z^5}{x^4}\ge1\)

Dấu "=" xay ra khi \(x=y=z=\frac{1}{3}\)

Áp dụng BĐT AM-GM:

P\(\le\Sigma\frac{x}{2\sqrt{x}}=\frac{x+y+z}{2}=1\)

Pmax=1 khi x=y=z=2/3.

\(VT=\sqrt[3]{1.1.\left(x+3y\right)}+\sqrt[3]{1.1.\left(y+3z\right)}+\sqrt[3]{1.1.\left(z+3x\right)}\)

\(VT\le\frac{1}{3}\left(1+1+x+3y\right)+\frac{1}{3}\left(1+1+y+3z\right)+\frac{1}{3}\left(1+1+z+3x\right)\)

\(VT\le\frac{1}{3}\left(6+4\left(x+y+z\right)\right)=3\)

Dấu "=" xảy ra khi \(x=y=z=\frac{1}{3}\)

\(A^3\le\left(x+3y+y+3z+z+3x\right)9=36\left(x+y+z\right)=27\Leftrightarrow A\le3\)