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\(\dfrac{x}{x+1}:\dfrac{x+2}{x+3}:\dfrac{x+3}{x+4}:\dfrac{x+4}{x+5}:\dfrac{x+5}{x+6}=\dfrac{x}{x+6}\)
Bài 1:
Vì $x+y+z=1$ nên:
\(Q=\frac{x}{x+\sqrt{x(x+y+z)+yz}}+\frac{y}{y+\sqrt{y(x+y+z)+xz}}+\frac{z}{z+\sqrt{z(x+y+z)+xy}}\)
\(Q=\frac{x}{x+\sqrt{(x+y)(x+z)}}+\frac{y}{y+\sqrt{(y+z)(y+x)}}+\frac{z}{z+\sqrt{(z+x)(z+y)}}\)
Áp dụng BĐT Bunhiacopxky:
\(\sqrt{(x+y)(x+z)}=\sqrt{(x+y)(z+x)}\geq \sqrt{(\sqrt{xz}+\sqrt{xy})^2}=\sqrt{xz}+\sqrt{xy}\)
\(\Rightarrow \frac{x}{x+\sqrt{(x+y)(x+z)}}\leq \frac{x}{x+\sqrt{xy}+\sqrt{xz}}=\frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}+\sqrt{z}}\)
Hoàn toàn tương tự với các phân thức còn lại và cộng theo vế suy ra:
\(Q\leq \frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}+\sqrt{z}}+ \frac{\sqrt{y}}{\sqrt{x}+\sqrt{y}+\sqrt{z}}+ \frac{\sqrt{z}}{\sqrt{x}+\sqrt{y}+\sqrt{z}}=1\)
Vậy $Q$ max bằng $1$
Dấu bằng xảy ra khi $x=y=z=\frac{1}{3}$
Bài 2:
Vì $x+y+z=1$ nên:
\(\text{VT}=\frac{1-x^2}{x(x+y+z)+yz}+\frac{1-y^2}{y(x+y+z)+xz}+\frac{1-z^2}{z(x+y+z)+xy}\)
\(\text{VT}=\frac{(x+y+z)^2-x^2}{(x+y)(x+z)}+\frac{(x+y+z)^2-y^2}{(y+z)(y+x)}+\frac{(x+y+z)^2-z^2}{(z+x)(z+y)}\)
\(\text{VT}=\frac{(y+z)[(x+y)+(x+z)]}{(x+y)(x+z)}+\frac{(x+z)[(y+z)+(y+x)]}{(y+z)(y+x)}+\frac{(x+y)[(z+x)+(z+y)]}{(z+x)(z+y)}\)
Áp dụng BĐT AM-GM:
\(\text{VT}\geq \frac{2(y+z)\sqrt{(x+y)(x+z)}}{(x+y)(x+z)}+\frac{2(x+z)\sqrt{(y+z)(y+x)}}{(y+z)(y+x)}+\frac{2(x+y)\sqrt{(z+x)(z+y)}}{(z+x)(z+y)}\)
\(\Leftrightarrow \text{VT}\geq 2\underbrace{\left(\frac{y+z}{\sqrt{(x+y)(x+z)}}+\frac{x+z}{\sqrt{(y+z)(y+x)}}+\frac{x+y}{\sqrt{(z+x)(z+y)}}\right)}_{M}\)
Tiếp tục AM-GM cho 3 số trong ngoặc lớn, suy ra \(M\geq 3\)
Do đó: \(\text{VT}\geq 2.3=6\) (đpcm)
Dấu bằng xảy ra khi $3x=3y=3z=1$
1/
A= \(\dfrac{2x+6}{\left(x+3\right)\left(x-2\right)}\) = 0 ;(ĐKXĐ : x ≠ -3; x ≠ 2)
⇔ A = \(\dfrac{2\left(x+3\right)}{\left(x+3\right)\left(x-2\right)}\) = 0
⇔ A = \(\dfrac{2}{x-2}\) = 0
⇒ x = 2 (loại) ⇒ pt vô nghiệm
\(=>\left(\left(\dfrac{1}{a}+\dfrac{1}{b}\right)+\dfrac{1}{c}\right)^2=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\)
Phân tích vế trái ta được ( hằng đẳng thức) :>
\(=\left(\dfrac{1}{a}+\dfrac{1}{b}\right)^2+\dfrac{2}{ac}+\dfrac{2}{bc}+\left(\dfrac{1}{c}\right)^2\)
\(=\dfrac{1}{a^2}+\dfrac{2}{ab}+\dfrac{1}{b^2}+\dfrac{2}{ac}+\dfrac{2}{bc}+\dfrac{1}{c^2}\)
\(=>\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2}{ac}+\dfrac{2}{ab}+\dfrac{2}{bc}=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}\)
\(=>2.\left(\dfrac{1}{ac}+\dfrac{1}{ab}+\dfrac{1}{bc}\right)=0\)
\(=>\dfrac{b}{abc}+\dfrac{c}{abc}+\dfrac{a}{abc}=0\)
\(=>a+b+c=0.abc=0\)
\(=>a+b=-c\)
\(=>-\left(a+b\right)=c\)
Thay vào ta có:
\(a^3+b^3+c^3=a^3+b^3-\left(a+b\right)^3\)
\(=-3a^2b-3ab^2=3\left(-a^2b-ab^2\right)⋮3\)
CHÚC BẠN HỌC TỐT NHA....
Cách khác:
Áp dụng BĐT AM-GM:
\(\frac{a}{b^2}+\frac{1}{a}\geq 2\sqrt{\frac{1}{b^2}}=\frac{2}{b}\)
\(\frac{b}{c^2}+\frac{1}{b}\geq 2\sqrt{\frac{1}{c^2}}=\frac{2}{c}\)
\(\frac{c}{a^2}+\frac{1}{c}\geq 2\sqrt{\frac{1}{a^2}}=\frac{2}{a}\)
Cộng theo vế và rút gọn:
\(\frac{a}{b^2}+\frac{b}{c^2}+\frac{c}{a^2}\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\) (đpcm)
Lời giải:
Ta có:
\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}\)
\(\Leftrightarrow \frac{1}{a}+\frac{1}{b}+\frac{1}{c}-\frac{1}{a+b+c}=0\)
\(\Leftrightarrow \frac{a+b}{ab}+\frac{a+b}{c(a+b+c)}=0\)
\(\Leftrightarrow (a+b)\left(\frac{1}{ab}+\frac{1}{c(a+b+c)}\right)=0\)
\(\Leftrightarrow \frac{(a+b)[c(a+b+c)+ab]}{abc(a+b+c)}=0\)
\(\Leftrightarrow (a+b)(b+c)(c+a)=0\)
Xét : \(A=\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}-\frac{1}{a^n+b^n+c^n}\)
\(A=\frac{a^n+b^n}{a^nb^n}+\frac{a^n+b^n}{c^n(a^n+b^n+c^n)}\)
\(A=(a^n+b^n)\left(\frac{1}{a^nb^n}+\frac{1}{c^n(a^n+b^n+c^n)}\right)\)
\(A=\frac{(a^n+b^n)[c^n(a^n+b^n+c^n)+a^nb^n]}{a^nb^nc^n(a^n+b^n+c^n)}\)
\(A=\frac{(a^n+b^n)(b^n+c^n)(c^n+a^n)}{a^nb^nc^n(a^n+b^n+c^n)}\)
Vì $n$ lẻ nên :
\((a^n+b^n)(b^n+c^n)(c^n+a^n)=(a+b)(b+c)(c+a)(a^{n-1}+....+b^{n-1})(b^{n-1}+..+c^{n-1})(c^{n-1}+...+a^{n-1})\)
\(=0\) do \((a+b)(b+c)(c+a)=0\)
Do đó: \(A=0\Leftrightarrow \frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}\)
C113
Ta có: \(\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = \dfrac{1}{{a + b + c}} \Longrightarrow \dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{{a + b + c}} - \dfrac{1}{c}\)
\(\begin{array}{l} \Longrightarrow \left( {a + b} \right)\left( {a + b + c} \right)c = abc - ab\left( {a + b + c} \right)\\ \Longrightarrow \left( {a + b} \right)\left( {b + c} \right)\left( {c + a} \right) = 0 \end{array}\)
............
C112:
a16 + a8b8 + b16
= a16 + 2a8b8 + b16 - a8b8
= (a8 + b8)2 - (a4b4)2
= (a8 + b8 - a4b4)(a8 + b8 + a4b4)