thử bài bất :D 

Ta có: \(\dfrac{1}{a^3\left(b+c\right)}+\dfrac{a}{2}+\dfrac{a}{2}+\dfrac{a}{2}+\dfrac{b+c}{4}\ge5\sqrt[5]{\dfrac{1}{a^3\left(b+c\right)}.\dfrac{a^3}{2^3}.\dfrac{\left(b+c\right)}{4}}=\dfrac{5}{2}\) ( AM-GM cho 5 số ) (*)

Hoàn toàn tương tự: 

\(\dfrac{1}{b^3\left(c+a\right)}+\dfrac{b}{2}+\dfrac{b}{2}+\dfrac{b}{2}+\dfrac{c+a}{4}\ge5\sqrt[5]{\dfrac{1}{b^3\left(c+a\right)}.\dfrac{b^3}{2^3}.\dfrac{\left(c+a\right)}{4}}=\dfrac{5}{2}\) (AM-GM cho 5 số) (**)

\(\dfrac{1}{c^3\left(a+b\right)}+\dfrac{c}{2}+\dfrac{c}{2}+\dfrac{c}{2}+\dfrac{a+b}{4}\ge5\sqrt[5]{\dfrac{1}{c^3\left(a+b\right)}.\dfrac{c^3}{2^3}.\dfrac{\left(a+b\right)}{4}}=\dfrac{5}{2}\) (AM-GM cho 5 số) (***)

Cộng (*),(**),(***) vế theo vế ta được:

\(P+\dfrac{3}{2}\left(a+b+c\right)+\dfrac{2\left(a+b+c\right)}{4}\ge\dfrac{15}{2}\) \(\Leftrightarrow P+2\left(a+b+c\right)\ge\dfrac{15}{2}\)

Mà: \(a+b+c\ge3\sqrt[3]{abc}=3\) ( AM-GM 3 số )

Từ đây: \(\Rightarrow P\ge\dfrac{15}{2}-2\left(a+b+c\right)=\dfrac{3}{2}\)

Dấu "=" xảy ra khi a=b=c=1

1. \(a^3+b^3+c^3+d^3=2\left(c^3-d^3\right)+c^3+d^3=3c^3-d^3\) :D 

Đặt A = \(\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\)

A = \(\left(1+\frac{a+b}{a}\right)\left(1+\frac{a+b}{b}\right)\)(Vì a + b = 1)

A = \(\left(2+\frac{b}{a}\right)\left(2+\frac{a}{b}\right)\)

A = \(4+\frac{2a}{b}+\frac{2b}{a}+1\)

A = \(5+2\left(\frac{a}{b}+\frac{b}{a}\right)\)

Vì a, b dương nên áp dụng BĐT Cô - si cho 2 số dương, ta được :

\(\frac{a}{b}+\frac{b}{a}\ge2\sqrt{\frac{ab}{ba}}\)

\(\Leftrightarrow\frac{a}{b}+\frac{b}{a}\ge2.1=2\)

\(\Leftrightarrow2\left(\frac{a}{b}+\frac{b}{a}\right)\ge4\)

\(\Leftrightarrow5+2\left(\frac{a}{b}+\frac{b}{a}\right)\ge4+5\)

\(\Leftrightarrow A\ge9\)

Dấu bằng xảy ra \(\Leftrightarrow\)a = b > 0

Vậy \(\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\ge9\)với a, b là các số dương và a + b = 1

Tớ quên. Dấu bằng xảy ra

\(\Leftrightarrow\hept{\begin{cases}a=b>0\\a+b=1\end{cases}}\)

\(\Leftrightarrow a=b=\frac{1}{2}\)

Một cách khác :))

Ta có : \(\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)=1+\frac{1}{b}+\frac{1}{a}+\frac{1}{ab}\)

Theo bđt Cauchy-Schwarz dạng Engel ta có : \(\frac{1}{b}+\frac{1}{a}\ge\frac{4}{a+b}=4\)(1)

Theo bđt AM-GM ta có : \(ab\le\left(\frac{a+b}{2}\right)^2=\left(\frac{1}{2}\right)^2=\frac{1}{4}\)=> \(\frac{1}{ab}\ge4\)(2)

Từ (1) và (2) => \(\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)=1+\frac{1}{b}+\frac{1}{a}+\frac{1}{ab}\ge1+4+4=9\left(đpcm\right)\)

Đẳng thức xảy ra <=> a = b = 1/2

Đặt \(P=\dfrac{1}{a^3\left(b+c\right)}+\dfrac{1}{b^3\left(c+a\right)}+\dfrac{1}{c^3\left(a+b\right)}\)

\(P=\dfrac{\left(abc\right)^2}{a^3\left(b+c\right)}+\dfrac{\left(abc\right)^2}{b^3\left(c+a\right)}+\dfrac{\left(abc\right)^2}{c^3\left(a+b\right)}\)

\(P=\dfrac{\left(bc\right)^2}{a\left(b+c\right)}+\dfrac{\left(ca\right)^2}{b\left(c+a\right)}+\dfrac{\left(ab\right)^2}{c\left(a+b\right)}\)

\(P\ge\dfrac{\left(bc+ca+ab\right)^2}{a\left(b+c\right)+b\left(c+a\right)+c\left(a+b\right)}\) (BĐT B.C.S)

\(=\dfrac{ab+bc+ca}{2}\) \(\ge\dfrac{3\sqrt[3]{abbcca}}{2}=\dfrac{3}{2}\) (do \(abc=1\)).

ĐTXR \(\Leftrightarrow a=b=c=1\)

Đặt \(a\left(1-b\right)=x;b\left(1-c\right)=y;c\left(1-a\right)=x\)

\(\Rightarrow1-\left(a+b+c\right)+ab+bc+ca=1-a\left(1-b\right)-b\left(1-c\right)-c\left(1-a\right)=1-x-y-z\)

BĐT cần c/m trở thành:

\(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\ge\dfrac{3}{1-x-y-z}\)

\(\Leftrightarrow\left(1-x-y-z\right)\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)-3\ge0\)

\(\Leftrightarrow\dfrac{1-x-y-z}{x}+\dfrac{1-x-y-z}{y}+\dfrac{1-x-y-z}{z}-3\ge0\)

\(\Leftrightarrow\dfrac{1-y-z}{x}+\dfrac{1-z-x}{y}+\dfrac{1-x-y}{z}-6\ge0\) (1)

Lại có: \(1-y-z=1-b\left(1-c\right)-c\left(1-a\right)=1-b-c+bc+ca=\left(1-b\right)\left(1-c\right)+ca\)

Nên (1) tương đương:

\(\dfrac{\left(1-b\right)\left(1-c\right)+ca}{a\left(1-b\right)}+\dfrac{\left(1-a\right)\left(1-c\right)+ab}{b\left(1-c\right)}+\dfrac{\left(1-a\right)\left(1-b\right)+bc}{c\left(1-a\right)}-6\ge0\)

\(\Leftrightarrow\dfrac{1-c}{a}+\dfrac{c}{1-b}+\dfrac{1-a}{b}+\dfrac{a}{1-c}+\dfrac{1-b}{c}+\dfrac{b}{1-a}\ge6\)

BĐT trên hiển nhiên đúng theo AM-GM do:

\(\dfrac{1-c}{a}+\dfrac{c}{1-b}+\dfrac{1-a}{b}+\dfrac{a}{1-c}+\dfrac{1-b}{c}+\dfrac{b}{1-a}\ge6\sqrt[6]{\dfrac{abc\left(1-a\right)\left(1-b\right)\left(1-c\right)}{abc\left(1-a\right)\left(1-b\right)\left(1-c\right)}}=6\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c=\dfrac{1}{2}\)

Ta có \(\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\ge9\)       (1)

\(\Leftrightarrow\frac{a+1}{a}.\frac{b+1}{b}\ge9\)

\(\Leftrightarrow ab+a+b+1\ge9ab\) (vì ab > 0)

\(\Leftrightarrow a+b+1\ge8ab\Leftrightarrow2\ge8ab\) (vì a + b = 1)

\(\Leftrightarrow1\ge4ab\Leftrightarrow\left(a+b\right)^2\ge4ab\)   (vì a + b = 1)

\(\Leftrightarrow\left(a-b\right)^2\ge0\)  (2)

Bất đẳng thức (2) đúng, mà các phép biến đổi trên tương đương, vậy bất đẳng thức (1) được chưng minh.

1+1/a= 1+ (a+b)/a = 2+b/a

tương tự: 1+1/b= 2+a/b

nhân 2 đa thức với nhau đc : 5+2a/b+2b/a=5+2(a/b+b/a)

áp dụng bđt cô si a/b+b/a >=2     =) 5+2(a/b+b/a)>=9 (dấu = xảy ra khi a-b=1/2)

\(LSH=\left(2+\frac{b}{a}\right)\left(2+\frac{a}{b}\right)=5+2\left(\frac{a}{b}+\frac{b}{a}\right)\ge5+2.2=RHS\)

a,b,c là các số dương nên \(\left(a+b+c\right)>=3\cdot\sqrt[3]{abc}\)

\(\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)>=3\cdot\sqrt[3]{\dfrac{1}{a}\cdot\dfrac{1}{b}\cdot\dfrac{1}{c}}\)

Do đó: \(\left(a+b+c\right)\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)>=3\cdot\sqrt[3]{abc}\cdot3\cdot\sqrt[3]{\dfrac{1}{a}\cdot\dfrac{1}{b}\cdot\dfrac{1}{c}}=9\cdot\sqrt[3]{a\cdot b\cdot c\cdot\dfrac{1}{a}\cdot\dfrac{1}{b}\cdot\dfrac{1}{c}}=9\)

Theo nguyên lý Dirichlet, trong 3 số a;b;c luôn có ít nhất 2 số cùng phía so với 1

Không mất tính tổng quát, giả sử đó là a và b

\(\Rightarrow\left(a-1\right)\left(b-1\right)\ge0\)

\(\Leftrightarrow ab+1\ge a+b\)

\(\Leftrightarrow2\left(ab+1\right)\ge\left(a+1\right)\left(b+1\right)\)

\(\Rightarrow\dfrac{2}{\left(a+1\right)\left(b+1\right)\left(c+1\right)}\ge\dfrac{2}{2\left(ab+1\right)\left(c+1\right)}=\dfrac{1}{\left(ab+1\right)\left(c+1\right)}=\dfrac{1}{\left(\dfrac{1}{c}+1\right)\left(c+1\right)}=\dfrac{c}{\left(c+1\right)^2}\)

Lại có:

\(\dfrac{1}{\left(\sqrt{ab}.\sqrt{\dfrac{a}{b}}+1.1\right)^2}+\dfrac{1}{\left(\sqrt{ab}.\sqrt{\dfrac{b}{a}}+1\right)^2}\ge\dfrac{1}{\left(ab+1\right)\left(\dfrac{a}{b}+1\right)}+\dfrac{1}{\left(ab+1\right)\left(\dfrac{b}{a}+1\right)}=\dfrac{1}{ab+1}\)

\(\Rightarrow P\ge\dfrac{1}{ab+1}+\dfrac{1}{\left(c+1\right)^2}+\dfrac{c}{\left(c+1\right)^2}=\dfrac{1}{\dfrac{1}{c}+1}+\dfrac{1}{\left(c+1\right)^2}+\dfrac{c}{\left(c+1\right)^2}\)

\(\Rightarrow P\ge\dfrac{c}{c+1}+\dfrac{c+1}{\left(c+1\right)^2}=\dfrac{c\left(c+1\right)+c+1}{\left(c+1\right)^2}=\dfrac{\left(c+1\right)^2}{\left(c+1\right)^2}=1\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c=1\)

Theo giả thiết kết hợp sử dụng BĐT AM - GM có:

\(\left(a+b-c\right)\left(\dfrac{1}{a}+\dfrac{1}{b}-\dfrac{1}{c}\right)=\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)+1-\left[c\left(a+b\right)+c\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\right]\)

\(\le\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)+1-2\sqrt{\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)}=\left[\sqrt{\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)}-1\right]^2\)

Suy ra \(\sqrt{\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)}-1\ge2\Leftrightarrow\sqrt{\dfrac{a}{b}+\dfrac{b}{a}+2}\ge3\)

\(\Leftrightarrow\dfrac{a}{b}+\dfrac{b}{a}\ge7\)

Khi đó, sử dụng BĐT Cauchy - Schwarz ta có:

\(\left(a^4+b^4+c^4\right)\left(\dfrac{1}{a^4}+\dfrac{1}{b^4}+\dfrac{1}{c^4}\right)\ge\left[\sqrt{\left(a^4+b^4\right)\left(\dfrac{1}{a^4}+\dfrac{1}{b^4}\right)}+1\right]^2\)

\(=\left(\dfrac{a^2}{b^2}+\dfrac{b^2}{a^2}+1\right)^2=\left[\left(\dfrac{a}{b}+\dfrac{b}{a}\right)^2-1\right]^2\ge\left(7^2-1\right)^2=2304\)

Đẳng thức xảy ra khi và chỉ khi \(ab=c^2\) và \(\dfrac{a}{b}+\dfrac{b}{a}=7\)

(a+b-c)(1/a+1/b-c)=(a+b)(1/a+1/b)+1-[c(a+b)+c(1/a+1/b)]<=(a+b)(1/a+1/b)+1-2căn (a+b)(1/a+1/b)

=[(căn (a+b)(1/a+1/b))-1]^2

=>\(\sqrt{\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)}-1>=2\)

=>\(\sqrt{\dfrac{a}{b}+\dfrac{b}{a}+2}>=3\)

=>a/b+b/a>=7

(a^4+b^4+c^4)(1/a^4+1/b^4+1/c^4)>=[căn ((a^4+b^4)(1/a^4+1/b^4))+1]^2

=(a^2/b^2+b^2/a^2+1)^2=[(a/b+b/a)^2-1]^2>=(7^2-1)^2=2304

=>ĐPCM