Từ đẳng thức đến bất đẳng thức

Bài viết này lấy nội dung từ bài báo “Từ những đẳng thức đẹp” của tác giả Đinh Ngọc Thuần. Bài viết thuộc kiến thức nâng cao bậc THCS các bạn học sinh lớp 8, lớp 9 có thể tham khảo để có thêm những kĩ năng chinh phục những bài tập khó trong các kì thi. Bạn đọc nên phân tích kĩ lời giải trình bày trong bài viết, nếu tự thực hiện được bài tập tương tự thì coi như đã nắm được ý tưởng của tác giả.
Nếu bạn gặp khó khăn hay thắc mắc gì hãy comment bên dưới, chúng tôi sẽ giúp đỡ các bạn
 
Bài toán: Cho 3 số thực $x,y,z$ thỏa mãn $xyz=1$. CMR
$\dfrac{1}{1+x+xy}+\dfrac{1}{1+y+yz}+\dfrac{1}{1+z+zx}=1.$
 
Chứng minh: Vì $xyz=1$ nên
$\dfrac{1}{1+y+yz}=\dfrac{x}{x+xy+xyz}=\dfrac{x}{x+xy+1};$
$\dfrac{1}{1+z+zx}=\dfrac{xy}{xy+xyz+x^2yz}=\dfrac{xy}{xy+1+x}.$
 
Từ đó $VT=\dfrac{1+x+xy}{1+x+xy}=1$, ta có điều phải chứng minh.
 
Bài tập tương tự:
 
1/ Cho $x,y,z$ là các số thực thỏa mãn $xyz=1$. Tính giá trị biểu thức
$A=\dfrac{1}{1+2x+3xy}+\dfrac{2}{2+3y+yz}+\dfrac{3}{3+x+2xz}$
$B=\dfrac{1}{1+x^2+x^2y^2}+\dfrac{1}{1+y^2+y^2z^2}+\dfrac{1}{1+z^2+z^2x^2}$
 
2/ Cho $x,y,z$ là các số thực thỏa mãn $xyz=2$. Chứng minh rằng
$\dfrac{x}{2+x+xy}+\dfrac{y}{1+y+yz}+\dfrac{2z}{2+2z+zx}=1.$
 
Vận dụng:
Bài 1: Cho 3 số thực dương $x,y,z$ thỏa mãn $xyz=1$. Chứng minh rằng
$\dfrac{1}{2x^2+y^2+3}+\dfrac{1}{2y^2+z^2+3}+\dfrac{1}{2z^2+x^2+3}\le\dfrac{1}{2}$
 
Chứng minh:
 
Ta có
$2x^2+y^2+3=(x-y)^2+(x-1)^2+2(1+x+xy)\ge 2(1+x+xy)$
$2y^2+z^2+3=(y-z)^2+(y-1)^2+2(1+y+yz)\ge 2(1+y+yz)$
 
Khi đó
$VT\le \dfrac{1}{2}\left(\dfrac{1}{1+x+xy}+\dfrac{1}{1+y+yz}+\dfrac{1}{1+z+zx}\right)\overset{(*)}{\mathop{=}}\dfrac{1}{2}.$
 
Biến đổi (*) là do áp dụng bài toán ban đầu.
 
Bình luận: Cách tách $2x^2+y^2+3=(x-y)^2+(x-1)^2+2(1+x+xy)$ có được là do ta dự đoán dấu “$=$” xảy ra khi $x=y=z=1$. Để hiểu rõ hơn ta xét thêm một bài toán nữa.
 
Bài 2: Cho 3 số thực dương $x,y,z$ thỏa mãn $xyz=2$. Chứng minh rằng
$\dfrac{x}{2x^2+y^2+5}+\dfrac{2y}{6y^2+z^2+6}+\dfrac{4z}{3z^2+4x^2+16}\le \dfrac{1}{2}$
 
Chứng minh: Dự đoán dấu “$=$” xảy ra khi $x=y=1$ và $z=2$, tương tự như bài trên ta sẽ tìm cách tách để xuất hiện $(x-y)^2$, $(x-1)^2$, $(y-1)^2$, $(z-2y)^2$, $(z-2x)^2$, $(z-2)^2$.
 
Ta có:
$2x^2+y^2+5=(x-y)^2+(x-1)^2+2(xy+x+2)\ge 2(xy+x+2)$
$6y^2+z^2+6=(z-2y)^2+2(y-1)^2+4(yz+y+1)\ge 4(yz+y+1)$
$3z^2+4x^2+16=(2x-z)^2+2(z-2)^2+4(xz +2z+2)\ge 4(xz +2z+2)$
 
Khi đó
$VT\le \dfrac{1}{2}\left(\dfrac{x}{2+x+xy}+\dfrac{y}{1+y+yz}+\dfrac{2z}{2+2z+zx}\right)=\dfrac{1}{2}.$
 
Bài tập tương tự:
 
1/ Cho các số dương $x,y,z$ thỏa mãn $xyz=1$. Tìm giá trị lớn nhất của biểu thức
$M=\dfrac{1}{(x+1)^2+y^2+1}+\dfrac{1}{(y+1)^2+z^2+1}+\dfrac{1}{(z+1)^2+x^2+1},$
$N=\dfrac{1}{5x^2+3y^2+4}+\dfrac{2}{5y^2+z^2+7}+\dfrac{3}{3x^2+2z^2+7}.$

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