\(F=\sqrt{-3x^2-6x+2}\left(Đk:-1-\sqrt{\dfrac{5}{3}}\le x\le\sqrt{\dfrac{5}{3}}-1\right)\)

\(=\sqrt{-\left(3x^2+6x+3\right)+5}\)

\(=\sqrt{-3\left(x+1\right)^2+5}\)

Vì \(-\left(x+1\right)^2\le0\forall x\)

\(\Rightarrow F\le\sqrt{5}\)

\(MaxF=\sqrt{5}\Leftrightarrow x=-1\)

\(y=\sqrt{2}sin\left(x+\dfrac{\text{π}}{4}\right)\)

\(=\sqrt{2}sinx.cos\dfrac{\text{π}}{4}+\sqrt{2}sin\dfrac{\text{π}}{4}.cosx\)

\(=\sqrt{2}sinx.\dfrac{\sqrt{2}}{2}+\sqrt{2}.\dfrac{\sqrt{2}}{2}+cosx\)

\(=sinx+cosx\)

Tập xác định của hàm số là 

\(\forall x\in D\) thì \(-x\in D\)

Ta có: \(f\left(-x\right)=sin\left(-x\right)+cos\left(-x\right)=-sinx+cosx\ne f\left(x\right)\)

Hàm y không chẵn cũng không lẻ

a) \(6=\sqrt[3]{6^3}=\sqrt{216}>\sqrt[3]{208}=2\sqrt[3]{26}\)

b) \(2\sqrt[3]{6}=\sqrt[3]{2^3.6}=\sqrt[3]{48}>\sqrt[3]{47}\)

a) \(\sqrt{2x}=12\left(đk:x\ge0\right)\)

\(2x=144\)

\(x=72\)

b) \(\sqrt{9x^2-6x}+1=10\)\(\left(Đk:x\le0;x\ge\dfrac{2}{3}\right)\)

\(\sqrt{9x^2-6x}=9\)

\(9x^2-6x=81\)

\(\left(3x-1\right)^2=82\)

\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{\sqrt{82}+1}{3}\\x=\dfrac{1-\sqrt{82}}{3}\end{matrix}\right.\)

c) \(x^2\sqrt{5}-\sqrt{125}=0\)

\(x^2\sqrt{5}=5\sqrt{5}\)

\(x^2=5\)

\(\Rightarrow\left[{}\begin{matrix}x=\sqrt{5}\\x=-\sqrt{5}\end{matrix}\right.\)

a) \(3\sqrt[3]{2}=\sqrt[3]{3^3.2}=\sqrt[3]{54}>\sqrt[3]{53}\)

b) \(22=\sqrt[3]{22^3}=\sqrt[3]{10648}>\sqrt[3]{10638}=3\sqrt[3]{394}\)

\(\sqrt[3]{\dfrac{343a^3b^6}{-216}}=\sqrt[3]{\dfrac{\left(7ab^2\right)^3}{-6^3}}=-\dfrac{7ab^2}{6}\)

\(A=\dfrac{10^{2024}+1}{10^{2023}+1}=\dfrac{10\left(10^{2023}+1\right)}{10^{2023}+1}-\dfrac{9}{10^{2023}+1}=1-\dfrac{9}{10^{2023}+1}\)

\(B=\dfrac{10^{2023}+1}{10^{2022}+1}=\dfrac{10\left(10^{2022}+1\right)}{10^{2022}+1}-\dfrac{9}{10^{2022}+1}=1-\dfrac{9}{10^{2022}+1}\)

Vì \(\dfrac{9}{10^{2023}+1}< \dfrac{9}{10^{2022}+1}\)

\(\Rightarrow A>B\)

\(D=2023-8x+2y+4xy-y^2-5x^2\)

\(=-\left(y^2+5x^2-4xy-2y+8x-2023\right)\)

\(=-\left(y^2-2.y.\left(2x+1\right)+\left(2x+1\right)^2-\left(2x+1\right)^2+5x^2+8x-2023\right)\)

\(=-\left[\left(y-2x-1\right)^2-4x^2-4x-1+5x^2+8x-2023\right]\)

\(=-\left[\left(y-2x-1\right)^2+x^2+4x-2024\right]\)

\(=-\left[\left(y-2x-1\right)^2+\left(x+2\right)^2\right]+2028\)

Vì \(-\left[\left(y-2x-1\right)^2+\left(x+2\right)^2\right]\le0\forall x,y\)

\(MaxD=2028\Leftrightarrow\left\{{}\begin{matrix}x=-2\\y=-3\end{matrix}\right.\)

\(\sqrt{2023+2025}=\sqrt{2.2024}\)

\(2\sqrt{2024}=\sqrt{4.2024}\)

\(\sqrt{2.2024}< \sqrt{4.2024}\)

=> \(\sqrt{2023+2025}< 2.\sqrt{2024}\)

a) \(A=x^2+6x+10\)

\(=\left(x+3\right)^2+1=\left(-103+3\right)^2+1=100^2+1=10001\)

b) \(B=x^3+6x^2+12x+12\)

\(=\left(x+2\right)^3+4=\left(8+2\right)^3+4=1004\)