Question

[tranthuylinh]

làm hộ mik câu 3 ( mik đang cần gấp ạ )

Answer 1

[Thanh Quân]

\(A=\dfrac{2\sqrt{x}+17}{\sqrt{x+5}}=\dfrac{2\sqrt{x}+10}{\sqrt{x}+5}+\dfrac{7}{\sqrt{x}+5}=2+\dfrac{7}{\sqrt{x}+5}\) 

Để \(A\) ∈ \(Z\) thì \(\dfrac{7}{\sqrt{x}+5}\) phải ∈ \(Z\)

=> \(\sqrt{x}+5\) ∈ \(Ư\left(7\right)=\left\{-7;-1;1;7\right\}\)

# Với \(\sqrt{x}+5=-7=>\sqrt{x}=-12\)(Loại)

#Với \(\sqrt{x}+5=-1=>\sqrt{x}=-6\)(Loại)

#Với \(\sqrt{x}+5=1=>\sqrt{x}=-4\left(Loại\right)\)

#Với \(\sqrt{x}+5=7=>\sqrt{x}=2< =>x=4\left(Nhận\right)\)

Vậy \(x=4\) thì \(A\)∈\(Z\)

Answer 2

[Thanh Quân]

\(\sqrt[3]{\dfrac{a^4}{b^2\left(a^2-ab+b^2\right)}}+\sqrt[3]{\dfrac{b^4}{c^2\left(b^2-bc+c^2\right)}}\sqrt[3]{\dfrac{c^4}{a^2\left(c^2-ac+b^2\right)}}\) \(\text{≥}3\)

\(Ta\) \(Có\) : \(\sqrt[3]{\dfrac{a^4}{b^2\left(a^2-ab+b^2\right)}}=\sqrt[3]{\dfrac{a^6}{ab.ab\left(a^2-ab+b^2\right)}}=\dfrac{a^2}{\sqrt[3]{ab.ab.\left(a^2-ab+b^2\right)}}\) 

\(Áp\) \(dụng\) \(bđt\) \(AM-GM\) 

\(\sqrt[3]{ab.ab\left(a^2-ab+b^2\right)}\text{≤}\)  \(\dfrac{ab+ab+a^2-ab+b^2}{3}\) 

\(=>\dfrac{a^2}{\sqrt[3]{ab.ab\left(a^2-ab+b^2\right)}}\) \(\text{≥}\) \(\dfrac{3a^2}{a^2+ab+b^2}\) \(Hay\) \(\sqrt[3]{\dfrac{a^4}{b^2\left(a^2-ab+b^2\right)}}\text{≥}\dfrac{3a^2}{a^2+ab+b^2}\)

Tương tự ta cũng có : 

\(\sqrt[3]{\dfrac{b^4}{c^2\left(b^2-bc+c^2\right)}}\text{≥}\dfrac{3b^2}{b^2+bc+c^2}\) 

\(\sqrt[3]{\dfrac{c^4}{a^2\left(c^2-ac+a^2\right)}}\text{≥}\dfrac{3c^2}{a^2+ac+c^2}\)

\(=>\text{​​}\text{​​}\)\(\sqrt[3]{\dfrac{a^4}{b^2\left(a^2-ab+b^2\right)}}+\sqrt[3]{\dfrac{b^4}{c^2\left(b^2-bc+c^2\right)}}\sqrt[3]{\dfrac{c^4}{a^2\left(c^2-ac+b^2\right)}}\)  \(\text{≥}\) \(3\left(\dfrac{a^2}{a^2+ab+b^2}+\dfrac{b^2}{b^2+bc+c^2}+\dfrac{c^2}{a^2+ac+c^2}\right)\) 

Cần c/m \(\left(\dfrac{a^2}{a^2+ab+b^2}+\dfrac{b^2}{b^2+bc+c^2}+\dfrac{c^2}{a^2+ac+c^2}\right)\) ≥ \(1\) 

Ta có : \(\dfrac{a^2}{a^2+ab+b^2}\text{≥}\dfrac{1}{3}\) 

\(< =>3a^2\text{≥}a^2+ab+b^2\) \(< =>2a^2-b\left(a+b\right)\text{≥}0\) (1)

Lại có : \(a^2\text{≥}-b\left(a+b\right)\) (2)

Từ (1) và (2) => \(\dfrac{a^2}{a^2+ab+b^2}\text{≥}\dfrac{1}{3}\)

Tương tự ta cũng có :

 \(\dfrac{b^2}{b^2+bc+c^2}\text{≥}\dfrac{1}{3}\) 

\(\dfrac{c^2}{a^2+ac+c^2}\text{≥}\dfrac{1}{3}\)

Do đó \(\dfrac{a^2}{a^2+ab+b^2}+\dfrac{b^2}{b^2+bc+c^2}+\dfrac{c^2}{a^2+ac+c^2}\text{≥}1\)

Suy ra :  \(\sqrt[3]{\dfrac{a^4}{b^2\left(a^2-ab+b^2\right)}}+\sqrt[3]{\dfrac{b^4}{c^2\left(b^2-bc+c^2\right)}}\sqrt[3]{\dfrac{c^4}{a^2\left(c^2-ac+b^2\right)}}\) \(\text{≥}\) \(3\) 

Đẳng thức xảy ra <=> \(a=b=c=1\)