Từ bđt Cauchy : \(a+b\ge2\sqrt{ab}\) ta suy ra được \(ab\le\frac{\left(a+b\right)^2}{4}\)

Áp dụng vào bài toán của bạn :

a/ \(y=\left(x+3\right)\left(5-x\right)\le\frac{\left(x+3+5-x\right)^2}{4}=...............\)

b/ Tương tự

c/ \(y=\left(x+3\right)\left(5-2x\right)=\frac{1}{2}.\left(2x+6\right)\left(5-2x\right)\le\frac{1}{2}.\frac{\left(2x+6+5-2x\right)^2}{4}=.............\)

d/ Tương tự

e/ \(y=\left(6x+3\right)\left(5-2x\right)=3\left(2x+1\right)\left(5-2x\right)\le3.\frac{\left(2x+1+5-2x\right)^2}{4}=.......\)

f/ Xét \(\frac{1}{y}=\frac{x^2+2}{x}=x+\frac{2}{x}\ge2\sqrt{x.\frac{2}{x}}=2\sqrt{2}\)

Suy ra \(y\le\frac{1}{2\sqrt{2}}\)

..........................

g/ Đặt \(t=x^2\) , \(t>0\) (Vì nếu t = 0 thì y = 0)

\(\frac{1}{y}=\frac{t^3+6t^2+12t+8}{t}=t^2+6t+\frac{8}{t}+12\)

\(=t^2+6t+\frac{8}{3t}+\frac{8}{3t}+\frac{8}{3t}+12\)

\(\ge5.\sqrt[5]{t^2.6t.\left(\frac{8}{3t}\right)^3}+12=.................\)

Từ đó đảo ngược y lại rồi đổi dấu \(\ge\) thành \(\le\)

Bài 1 :

Ta có : \(\dfrac{3x+5}{2}-1\le\dfrac{x+2}{3}+x\)

\(\Leftrightarrow\dfrac{3x+5}{2}-1-\dfrac{x+2}{3}-x\le0\)

\(\Leftrightarrow\dfrac{3\left(3x+5\right)-6-2\left(x+2\right)-6x}{6}\le0\)

\(\Leftrightarrow9x+15-6-2x-4-6x\le0\)

\(\Leftrightarrow x\le-5\)

Mà \(\left\{{}\begin{matrix}x\in Z\\x>-10\end{matrix}\right.\)

Vậy \(x\in\left\{-5;-6;-7;-8;-9\right\}\)

b3\(\Leftrightarrow2x^2+5x-3-3x+1\le x^2+2x-3+x^2-5\\ \Leftrightarrow0.x\le-6\Leftrightarrow x\in\varnothing\)

1.

ĐK: \(x\ne7;x\ne-1;x\ne3\)

\(\dfrac{2x-5}{x^2-6x-7}\le\dfrac{1}{x-3}\left(1\right)\)

TH1: \(x< -1\)

\(\left(1\right)\Leftrightarrow\left(2x-5\right)\left(x-3\right)\ge x^2-6x-7\)

\(\Leftrightarrow2x^2-11x+15\ge x^2-6x-7\)

\(\Leftrightarrow x^2-5x+22\ge0\)

\(\Leftrightarrow\) Bất phương trình đúng với mọi \(x< -1\)

TH2: \(-1< x< 3\)

\(\left(1\right)\Leftrightarrow\left(3-x\right)\left(2x-5\right)\ge\left(7-x\right)\left(x+1\right)\)

\(\Leftrightarrow-2x^2+11x-15\ge-x^2+6x+7\)

\(\Leftrightarrow-x^2+5x-22\ge0\)

\(\Rightarrow\) vô nghiệm

TH3: \(3< x< 7\)

Khi đó \(\dfrac{2x-5}{x^2-6x-7}\le0\); \(\dfrac{1}{x-3}>0\)

\(\Rightarrow\) Bất phương trình đúng với mọi \(3< x< 7\)

TH4: \(x>7\)

\(\left(1\right)\Leftrightarrow\left(2x-5\right)\left(x-3\right)\le x^2-6x-7\)

\(\Leftrightarrow2x^2-11x+15\le x^2-6x-7\)

\(\Leftrightarrow x^2-5x+22\le0\)

\(\Rightarrow\) vô nghiệm

Vậy ...

Các bài kia tương tự, chứ giải ra mệt lắm.

a/ \(y=\left(x+3\right)\left(5-x\right)\le\frac{1}{4}\left(x+3+5-x\right)^2=16\)

Dấu "=" xảy ra khi \(x+3=5-x\Leftrightarrow x=1\)

b/ \(y=x\left(6-x\right)\le\frac{1}{4}\left(x+6-x\right)^2=9\)

\("="\Leftrightarrow x=3\)

c/ \(y=\frac{1}{2}\left(2x+6\right)\left(5-2x\right)\le\frac{1}{8}\left(2x+6+5-2x\right)^2=\frac{121}{8}\)

\("="\Leftrightarrow x=-\frac{1}{4}\)

d/ \(y=\frac{1}{2}\left(2x+5\right)\left(10-2x\right)\le\frac{1}{8}\left(2x+5+10-2x\right)^2=\frac{225}{8}\)

\("="\Leftrightarrow x=\frac{5}{4}\)

e/ \(y=3\left(2x+1\right)\left(5-2x\right)\le\frac{3}{4}\left(2x+1+5-2x\right)^2=27\)

\("="\Leftrightarrow x=1\)

f/ \(\frac{x}{x^2+2}\le\frac{x}{2\sqrt{x^2.2}}=\frac{1}{2\sqrt{2}}\)

\("="\Leftrightarrow x=\sqrt{2}\)

g/ \(y=\frac{x^2}{\left(x^2+\frac{3}{2}+\frac{3}{2}\right)^3}\le\frac{x^2}{\left(3\sqrt[3]{\frac{9}{4}x^2}\right)^3}=\frac{4}{243}\)

\("="\Leftrightarrow x^2=\frac{3}{2}\Leftrightarrow x=\pm\sqrt{\frac{3}{2}}\)

a, \(\left|x+2\right|+\left|-2x+1\right|\le x+1\left(1\right)\)

TH1: \(x\le-2\)

\(\Rightarrow x+1\le-1< \left|x+2\right|+\left|-2x+1\right|\)

\(\Rightarrow\) vô nghiệm

TH2: \(-2< x\le\dfrac{1}{2}\)

\(\left(1\right)\Leftrightarrow x+2-2x+1\le x+1\)

\(\Leftrightarrow x\ge1\)

\(\Rightarrow x\in\left[1;\dfrac{1}{2}\right]\)

TH3: \(x>\dfrac{1}{2}\)

\(\left(1\right)\Leftrightarrow x+2+2x-1\le x+1\)

\(\Leftrightarrow x\le0\)

\(\Rightarrow\) vô nghiệm

Vậy \(x\in\left[1;\dfrac{1}{2}\right]\)

b, \(\left|x+2\right|-\left|x-1\right|< x-\dfrac{3}{2}\left(2\right)\)

TH1: \(x\le-2\)

\(\left(2\right)\Leftrightarrow-x-2+x-1< x-\dfrac{3}{2}\)

\(\Leftrightarrow x>-\dfrac{3}{2}\)

\(\Rightarrow\) vô nghiệm

TH2: \(-2< x\le1\)

\(\left(2\right)\Leftrightarrow x+2+x-1< x-\dfrac{3}{2}\)

\(\Leftrightarrow x< -\dfrac{5}{2}\)

\(\Rightarrow\) vô nghiệm

TH3: \(x>1\)

\(\left(2\right)\Leftrightarrow x+2-x+1< x-\dfrac{3}{2}\)

\(\Leftrightarrow x>\dfrac{9}{2}\)

\(\Rightarrow x\in\left(\dfrac{9}{2};+\infty\right)\)

Vậy \(x\in\left(\dfrac{9}{2};+\infty\right)\)

c, Tương tự a,b

d, ĐK: \(x\ne-2;x\ne1\)

\(\left|\dfrac{-5}{x+2}\right|< \left|\dfrac{10}{x-1}\right|\)

\(\Leftrightarrow\dfrac{1}{\left|x+2\right|}< \dfrac{2}{\left|x-1\right|}\)

\(\Leftrightarrow2\left|x+2\right|>\left|x-1\right|\)

\(\Leftrightarrow4\left(x+2\right)^2>\left(x-1\right)^2\)

\(\Leftrightarrow4\left(x^2+4x+4\right)>x^2-2x+1\)

\(\Leftrightarrow3x^2+18x+15>0\)

\(\Leftrightarrow...\)

e, ĐK: \(x\ne-1\)

\(\left|\dfrac{2-3\left|x\right|}{1+x}\right|\le1\)

\(\Leftrightarrow\left|2-3\left|x\right|\right|\le\left|x+1\right|\)

\(\Leftrightarrow\left(2-3\left|x\right|\right)^2\le\left(x+1\right)^2\)

\(\Leftrightarrow4+9x^2-12\left|x\right|\le x^2+2x+1\)

\(\Leftrightarrow8x^2-12\left|x\right|-2x+3\le0\)

Đến đây dễ rồi, xét hai trường hợp để phá dấu giá trị tuyệt đối rồi đối chiếu điêì kiện.

ĐK: \(-5\le x\le3\)

\(\sqrt{\left(x+5\right)\left(3-x\right)}\le x^2+2x+m\)

\(\Leftrightarrow-x^2-2x+15+\sqrt{-x^2-2x+15}-15\le m\)

Đặt \(\sqrt{-x^2-2x+15}=t\left(0\le t\le4\right)\)

Bất phương trình đã cho tương đương:

\(\Leftrightarrow f\left(t\right)=t^2+t-15\le m\)

Yêu cầu bài toán thỏa mãn khi \(m\ge maxf\left(t\right)=f\left(4\right)=5\)

Vậy \(m\ge5\)

A

Áp dụng BĐT cosi ta có 

\(\sqrt{\left(2x-1\right).1}\le\frac{2x-1+1}{2}=x\)

\(x\sqrt{5-4x^2}\le\frac{x^2+5-4x^2}{2}=\frac{-3x^2+5}{2}\)

Khi đó 

\(A\le3x+\frac{-3x^2+5}{2}=\frac{-3x^2+6x+5}{2}=\frac{-3\left(x-1\right)^2}{2}+4\le4\)

MaxA=4 khi \(\hept{\begin{cases}2x-1=1\\x^2=5-4x^2\\x=1\end{cases}\Rightarrow}x=1\)

B

Áp dụng BĐT cosi ta có :

\(x^2+y^2+z^2\ge\frac{1}{3}\left(x+y+z\right)^2\)

=> \(x+y+z\le\sqrt{3\left(x^2+y^2+z^2\right)}\)

=> \(B\le\frac{xyz.\left(\sqrt{3\left(x^2+y^2+z^2\right)}+\sqrt{x^2+y^2+z^2}\right)}{\left(x^2+y^2+z^2\right)\left(xy+yz+xz\right)}=\frac{xyz.\left(\sqrt{3}+1\right)}{\left(xy+yz+xz\right)\sqrt{x^2+y^2+z^2}}\)

Lại có \(x^2+y^2+z^2\ge3\sqrt[3]{x^2y^2z^2}\); \(xy+yz+xz\ge3\sqrt[3]{x^2y^2z^2}\)

=> \(\sqrt{x^2+y^2+z^2}\left(xy+yz+xz\right)\ge3\sqrt[3]{x^2y^2z^2}.\sqrt{3\sqrt[3]{x^2y^2z^2}}=3\sqrt{3}.xyz\)

=> \(B\le\frac{\sqrt{3}+1}{3\sqrt{3}}=\frac{3+\sqrt{3}}{9}\)

\(MaxB=\frac{3+\sqrt{3}}{9}\)khi x=y=z

\(\left(x-4\right).\left(x+4\right)\ge\left(x+3\right)^2+5\)

\(\Rightarrow x^2-16\ge x^2+6x+9+5\)

\(\Rightarrow x^2-16\ge x^2+6x+14\)

\(\Rightarrow-30\ge6x\Rightarrow-5\ge x\)

Vậy...