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Đặt A = \(\dfrac{a-b}{1+c^2}+\dfrac{b-c}{1+a^2}+\dfrac{c-a}{1+b^2}=0\)
= \(\dfrac{a-b}{c^2+ab+bc+ca}+\dfrac{b-c}{a^2+ab+bc+ca}+\dfrac{c-a}{b^2+ab+bc+ca}\)
= \(\dfrac{a-b}{\left(c+a\right)\left(c+b\right)}+\dfrac{b-c}{\left(a+b\right)\left(c+a\right)}+\dfrac{c-a}{\left(a+b\right)\left(b+c\right)}\)
= \(\dfrac{\left(a-b\right)\left(a+b\right)+\left(b-c\right)\left(b+c\right)+\left(c+a\right)\left(c-a\right)}{\left(c+a\right)\left(b+c\right)\left(a+b\right)}\)
= \(\dfrac{a^2-b^2+b^2-c^2+c^2-a^2}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=0\)
\(\dfrac{a-b}{1+c^2}+\dfrac{b-c}{1+a^2}+\dfrac{c-a}{1+b^2}\)
\(=\dfrac{a-b}{ab+bc+ca+c^2}+\dfrac{b-c}{ab+bc+ca+a^2}+\dfrac{c-a}{ab+bc+ca+b^2}\)
\(=\dfrac{a-b}{\left(c+a\right)\left(c+b\right)}+\dfrac{b-c}{\left(a+b\right)\left(a+c\right)}+\dfrac{c-a}{\left(b+a\right)\left(b+c\right)}\)
\(=\dfrac{\left(a-b\right)\left(a+b\right)+\left(b-c\right)\left(b+c\right)+\left(c-a\right)\left(c+a\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)
\(=\dfrac{a^2-b^2+b^2-c^2+c^2-a^2}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=0\)
Lời giải:
$\text{VT}=\frac{a(a+b+c)+bc}{b+c}+\frac{b(a+b+c)+ac}{a+c}+\frac{c(a+b+c)+ab}{a+b}$
$=\frac{(a+b)(a+c)}{b+c}+\frac{(b+a)(b+c)}{a+c}+\frac{(c+a)(c+b)}{a+b}$
Áp dụng BĐT AM-GM:
$\frac{(a+b)(a+c)}{b+c}+\frac{(b+a)(b+c)}{a+c}\geq 2\sqrt{(a+b)^2}=2(a+b)$
$\frac{(b+c)(b+a)}{a+c}+\frac{(c+a)(c+b)}{a+b}\geq 2\sqrt{(b+c)^2}=2(b+c)$
$\frac{(a+b)(a+c)}{b+c}+\frac{(c+a)(c+b)}{a+b}\geq 2\sqrt{(c+a)^2}=2(a+c)$
Cộng các BĐT trên theo vế và thu gọn:
$\text{VT}\geq 2(a+b+c)=2$
Ta có đpcm
Dấu "=" xảy ra khi $a=b=c=\frac{1}{3}$
1.
- Với \(a+b\ge4\Rightarrow A\le0\)
- Với \(a+b< 4\Rightarrow4-a-b>0\)
\(\Rightarrow A=\dfrac{a}{2}.\dfrac{a}{2}.b.\left(4-a-b\right)\)
\(\Rightarrow A\le\dfrac{1}{64}\left(\dfrac{a}{2}+\dfrac{a}{2}+b+4-a-b\right)^4=4\)
\(A_{max}=4\) khi \(\left(a;b\right)=\left(2;1\right)\)
2.
\(P=a+\dfrac{1}{2}.a.2b\left(1+2c\right)\le a+\dfrac{a}{8}\left(2b+1+2c\right)^2\)
\(P\le a+\dfrac{a}{8}\left(7-2a\right)^2=\dfrac{1}{8}\left(4a^3-28a^2+57a-36\right)+\dfrac{9}{2}\)
\(P\le\dfrac{1}{8}\left(a-4\right)\left(2a-3\right)^2+\dfrac{9}{2}\le\dfrac{9}{2}\)
Dấu "=" xảy ra khi \(\left(a;b;c\right)=\left(\dfrac{3}{2};1;\dfrac{1}{2}\right)\)
Câu 3 bạn xem lại đề, mình có thể chắc chắn với bạn là đề sai
Ví dụ bạn cho \(x=98,y=100\) thì vế trái chỉ lớn hơn 8 một chút
Đề đúng phải là: \(\left(x+y\right)\left(\dfrac{1}{x}+\dfrac{1}{y}\right)+\dfrac{16xy}{\left(x-y\right)^2}\ge12\)
Sửa \(\le\) thành \(\ge\) nha bạn
\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1\Leftrightarrow ab+bc+ca=abc\)
Ta có \(\dfrac{a^2}{a+bc}=\dfrac{a^3}{a^2+abc}=\dfrac{a^3}{a^2+ab+bc+ca}=\dfrac{a^3}{\left(a+b\right)\left(a+c\right)}\)
Tương tự: \(\left\{{}\begin{matrix}\dfrac{b^2}{b+ca}=\dfrac{b^3}{\left(b+a\right)\left(b+c\right)}\\\dfrac{c^2}{c+ba}=\dfrac{c^3}{\left(c+b\right)\left(c+a\right)}\end{matrix}\right.\)
Áp dụng BĐT cosi:
\(\dfrac{a^3}{\left(a+b\right)\left(a+c\right)}+\dfrac{a+b}{8}+\dfrac{a+c}{8}\ge3\sqrt[3]{\dfrac{a^3}{64}}=\dfrac{3}{4}a\)
\(\dfrac{b^3}{\left(b+a\right)\left(b+c\right)}+\dfrac{a+b}{8}+\dfrac{b+c}{8}\ge3\sqrt[3]{\dfrac{b^3}{64}}=\dfrac{3}{4}b\)
\(\dfrac{c^3}{\left(c+b\right)\left(c+a\right)}+\dfrac{b+c}{8}+\dfrac{a+c}{8}\ge3\sqrt[3]{\dfrac{c^3}{64}}=\dfrac{3}{4}c\)
Cộng VTV:
\(\Leftrightarrow VT+\dfrac{a+b}{8}+\dfrac{a+c}{8}+\dfrac{b+c}{8}\ge\dfrac{3}{4}\left(a+b+c\right)\\ \Leftrightarrow VT\ge\dfrac{3\left(a+b+c\right)}{4}-\dfrac{2\left(a+b+c\right)}{8}\\ \Leftrightarrow VT\ge\dfrac{a+b+c}{4}\)
Dấu \("="\Leftrightarrow a=b=c=3\)
Do \(abc=1\Rightarrow\) đặt \(\left(a;b;c\right)=\left(\dfrac{x}{y};\dfrac{y}{z};\dfrac{z}{x}\right)\)
\(VT=\dfrac{xz}{y\left(x+z\right)}+\dfrac{xy}{z\left(x+y\right)}+\dfrac{yz}{x\left(y+z\right)}=\dfrac{\left(xz\right)^2}{xyz\left(x+z\right)}+\dfrac{\left(xy\right)^2}{xyz\left(x+y\right)}+\dfrac{\left(yz\right)^2}{xyz\left(y+z\right)}\)
\(VT\ge\dfrac{\left(xy+yz+zx\right)^2}{2xyz\left(x+y+z\right)}\ge\dfrac{3xyz\left(x+y+z\right)}{2xyz\left(x+y+z\right)}=\dfrac{3}{2}\)
Dấu "=" xảy ra khi \(x=y=z\) hay \(a=b=c=1\)
Đề bài bị nhầm phải ko bạn.
Ta đặt P=\(\dfrac{b^3}{a}+\dfrac{a^3}{c}+\dfrac{c^3}{b}\) .Ta cần chứng minh P\(\ge3\)\(\dfrac{b^3}{a}+ab\ge2b^2;\dfrac{a^3}{c}+ac\ge2a^2;\dfrac{c^3}{b}+bc\ge2c^2\Rightarrow\dfrac{b^3}{a}+\dfrac{a^3}{c}+\dfrac{c^3}{b}\ge2a^2+2b^2+2c^2-ab-ca-bc\ge ab+bc+ca\Rightarrow2\cdot P\ge2ab+2bc+2ca\left(1\right)\) \(\dfrac{b^3}{a}+a+1\ge3b;\dfrac{a^3}{c}+c+1\ge3a;\dfrac{c^3}{b}+b+1\ge3c\Rightarrow\dfrac{b^3}{a}+\dfrac{a^3}{c}+\dfrac{c^3}{b}\ge3a+3b+3c-3-a-b-c=2a+2b+2c-3\left(2\right)\) Cộng từng vế của 2 bđt (1) và (2) ta được:
\(\Rightarrow3\cdot\left(\dfrac{b^3}{a}+\dfrac{a^3}{c}+\dfrac{c^3}{b}\right)\ge2\left(a+b+c+ab+bc+ca\right)-3=12-3=9\Rightarrow3P\ge9\Rightarrow P\ge3\) Dấu = xảy ra \(\Leftrightarrow a=b=c=1\)
\(\Leftrightarrow c-a=\dfrac{b}{a}-\dfrac{1}{b}=\dfrac{b^2-a}{ab}\)
\(\Rightarrow b^2-a=ab\left(c-a\right)\Rightarrow b^2=a\left[b\left(c-a\right)+1\right]\)
\(\Rightarrow b^2⋮b\left(c-a\right)+1\) (1)
Nếu \(b\left(c-a\right)+1\ne1\) , do b và \(b\left(c-a\right)+1\) nguyên tố cùng nhau
\(\Rightarrow b⋮̸b\left(c-a\right)+1\Rightarrow b^2⋮̸b\left(c-a\right)+1\) trái với (1)
\(\Rightarrow b\left(c-a\right)+1=1\Rightarrow c=a\)
\(\Rightarrow b^2=a\Rightarrow ab=b^3\) là lập phương 1 số tự nhiên