chứng minh các bất đẳng thức sau:

a) a²b+\(\frac{1}{b}\ge2a,\left(\forall a,b>0\right)\)

b) (a+b)(ab+1)≥4ab,(∀a,b>0)

c) (a+b)(a+2)(b+2)≥16ab, (∀a,b>0)

d) (1+\(\frac{a}{b}\))\(\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right)\ge8,\left(\forall a.b,c>0\right)\)

1/ cho \(\frac{a}{b}=\frac{c}{d}\) chứng minh rằng:

a) \(\frac{a.b}{c.d}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}\)

b) \(\frac{a.d}{c.b}=\frac{\left(a+b\right).\left(a-b\right)}{\left(c+d\right).\left(c-d\right)}\)

2/ cho a.b=c² chứng minh: \(\frac{a}{b}=\frac{\left(2.a+3.c\right)^2}{\left(2.c\right)+\left(3.b\right)^2}\)

chứng minh bất đẳng thức :

\(\left(a+b\right)^2\le2\left(a^2+b^2\right)\)

Chứng minh rằng với \(\forall a;b;c\)ta có: 

   \(\left(a^2+2\right)\left(b^2+2\right)\left(c^2+2\right)\ge3\left(a+b+c\right)^2\)

Cho \(f\left(x\right)=ax^2+bx+c\) thỏa mãn \(|f\left(x\right)|\le1,\forall|x|\le1\). Chứng minh rằng \(|f\left(x\right)|\le7,\forall|x|\le2\)

Cho \(a,b,c>0\). Chứng minh \(\dfrac{\left(1+a^2b\right)\left(1+b^2\right)}{\left(a^2-a+1\right)\left(b^3+1\right)}\le2\)

1/ cho \(\frac{a}{b}=\frac{c}{d}\)chứng minh rằng:

a) \(\frac{a.b}{c.d}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}\)

b)\(\frac{a,d}{c.b}=\frac{\left(a+b\right).\left(a-b\right)}{\left(c+d\right).\left(c-d\right)}\)

2/ cho \(a.b=c^2\)chứng minh : \(\frac{a}{b}=\frac{\left(2a+3c\right)^2}{\left(2c+3b\right)^2}\)

cho \(f\left(x\right)=ax^2+bx+c\) thỏa mãn |f(x)| ≤ 1 \(\forall x\in\left[-1;1\right]\). Chứng minh rằng \(\left|a\right|+\left|b\right|+\left|c\right|\le4\)

Chứng minh rằng \(\forall\) STN n ta có:

a) \(\left(7^n+1\right).\left(7^n+2\right)⋮3\)

b) \(n^2+n+6⋮̸4\)