\(a)\) Giả sử \(\left|x\right|+\left|y\right|\ge\left|x+y\right|\)

\(\Leftrightarrow\)\(\left(\left|x\right|+\left|y\right|\right)^2\ge\left|x+y\right|^2\)

\(\Leftrightarrow\)\(\left|x\right|^2+2\left|xy\right|+\left|y\right|^2\ge\left(x+y\right)^2\)

\(\Leftrightarrow\)\(x^2+2\left|xy\right|+y^2\ge x^2+2xy+y^2\)

\(\Leftrightarrow\)\(2\left|xy\right|\ge2xy\)

\(\Leftrightarrow\)\(\left|xy\right|\ge xy\) ( luôn đúng ) 

\(b)\) Giả sử \(\left|x\right|-\left|y\right|\le\left|x-y\right|\)

\(\Leftrightarrow\)\(\left(\left|x\right|-\left|y\right|\right)^2\le\left|x-y\right|^2\)

\(\Leftrightarrow\)\(\left|x\right|^2-2\left|xy\right|+\left|y\right|^2\le\left(x-y\right)^2\)

\(\Leftrightarrow\)\(x^2-2\left|xy\right|+y^2\le x^2-2xy+y^2\)

\(\Leftrightarrow\)\(-2\left|xy\right|\le-2xy\)

\(\Leftrightarrow\)\(\left|xy\right|\ge xy\) ( luôn đúng ) 

Chúc bạn học tốt ~ 

Ta có:

\(3^{n+2}-2^{n+2}+3^n-2^n\)

\(=\left(3^{n+2}+3^n\right)-\left(2^{n+2}+2^n\right)\)

\(=\left(3^n.3^2+3^n.1\right)-\left(2^n.2^2+2^n.1\right)\)

\(=3^n\left(3^2+1\right)-2^n\left(2^2+1\right)\)

\(=3^n\left(9+1\right)-2^{n-1}.2^1\left(4+1\right)\)

\(=3^n.10-2^{n-1}.2.5\)

\(=3^n.10-2^{n-1}.10\)

\(=\left(3^n-2^{n-1}\right).10\text{⋮}10\)

Đặt \(ab+4=m^2\left(m\in N^+\right)\)

\(\Rightarrow ab=m^2-4\)

\(\Leftrightarrow ab=\left(m-2\right)\left(m+2\right)\)

\(\Leftrightarrow b=\frac{\left(m+2\right)\left(m-2\right)}{a}\)

=> dpcm

Gọi 3 số tự nhiên cần tìm là a-2,a,a+2 
Ta có:(a-2)a+192=a(a+2) 
<->a^2-2a+192=a^2+2a 
<->192=a^2+2a-a^2+2a 
<->192=4a 
<->a=48 
-->a-2=46 
a+2=50 
Vây 3 số chẵn cần tìm là 46,48,50