Sao không nói x , y , z thuộc N cho nhanh bạn

Áp dụng BĐT Bunhiacopski ta có:

\(\sqrt{x^2+\frac{1}{x^2}}=\frac{1}{\sqrt{17}}\sqrt{\left(x^2+\frac{1}{x^2}\right)\left(4^2+1^2\right)}\ge\frac{1}{\sqrt{17}}\left(4x+\frac{1}{x}\right)\)

Tương tự:

\(\sqrt{y^2+\frac{1}{y^2}}\ge\frac{1}{\sqrt{17}}\left(4y+\frac{1}{y}\right)\)

Cộng lại ta được:

\(\sqrt{x^2+\frac{1}{x^2}}+\sqrt{y^2+\frac{1}{y^2}}\ge\frac{1}{\sqrt{17}}\left(4x+4y+\frac{1}{x}+\frac{1}{y}\right)\)

\(\ge\frac{1}{\sqrt{17}}\left[4\left(x+y\right)+\frac{4}{x+y}\right]=\frac{1}{\sqrt{17}}\left(16+1\right)=\sqrt{17}\)

Dấu "=" xảy ra tại x=y=2

A.2

......

Chúc học tốt

lớp 9 làm quen không bạn ^^

Chứng minh : a³ + b³ + c³ = 3abc \(\Rightarrow\orbr{\begin{cases}a+b+c=0\left(tm\right)\\a=b=c\left(loai\right)\end{cases}}\)

Rút gọn P

\(P=\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}=\frac{ab\left(a-b\right)+bc\left(b-c\right)+ac\left(c-a\right)}{abc}\)

Xét : ab(a-b) + bc(b-c) + ac(c-a) = ab[-(b-c)-(c-a)] + bc(b-c) + ac(c-a)

= (b-c)(bc-ab) + (c-a)(ac-ab) = b(b-c)(c-a) + a(c-a)(c-b) = (c-a)(c-b)(a-b)

\(\Rightarrow P=\frac{\left(c-a\right)\left(c-b\right)\left(a-b\right)}{abc}\)

Rút gọn Q

Đặt a - b = z ; b-c = x ; c - a = y

\(\Rightarrow\)x- y = a + b - 2c = -c - 2c = -3c              ( do a + b + c = 0 )

y - z = -3a ; z - x = -3b

\(\Rightarrow\)\(-3Q=\frac{\left(y-z\right)}{x}+\frac{\left(z-x\right)}{y}+\frac{\left(x-y\right)}{z}\)

Làm tương tự như rút gọn P, ta có : 

\(-3Q=\frac{\left(x-y\right)\left(z-y\right)\left(z-x\right)}{xyz}=\frac{-\left(-3a\right)\left(-3b\right)\left(-3c\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=\frac{27abc}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=\frac{-27abc}{\left(a-b\right)\left(c-b\right)\left(c-a\right)}\)

\(\Rightarrow Q=\frac{9abc}{\left(a-b\right)\left(c-b\right)\left(c-a\right)}\)

\(\Rightarrow PQ=9\)

\(T=\sqrt{x^2-x+2}+\sqrt{x^2+x+2}\)

\(T^2=x^2-x+2+x^2+x+2+2\sqrt{\left(x^2-x+2\right)\left(x^2+x+2\right)}\)

\(T^2=2x^2+4+2\sqrt{\left(x^2+2\right)^2-x^2}\)

\(T^2=2x^2+4+2\sqrt{x^4+4x^2+4-x^2}\)

\(T^2=2x^2+4+2\sqrt{x^4+3x^2+4}\)

Nhận xét : \(2x^2\ge0\forall x\)

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

Có : \(x^2\ge0,x^2+3\ge0\forall x\)nên 

\(\Rightarrow x^2\left(x^2+3\right)\ge0\forall x\)

Cho nên \(x^4+3x^2+4\ge4\)

Vậy \(T^2=2x^2+4+2\sqrt{x^4+3x^2+4}\ge4+2\sqrt{4}=4+4=8\)

Do \(T^2\ge8\)nên :

\(\Rightarrow A\ge2\sqrt{2}\)

Dấu " = " xảy ra \(\Leftrightarrow x=0\)