Áp dụng BĐT Bernoulli ta có:
\(\left(\frac{2x}{x+y}\right)^n=\left(1+\frac{x-y}{x+y}\right)^n\ge1+\frac{n\left(x-y\right)}{x+y}\)
\(\left(\frac{2y}{x+y}\right)^n=\left(1-\frac{x-y}{x+y}\right)^n\ge1-\frac{n\left(x-y\right)}{x+y}\)
Cộng theo vế 2 BĐT trên ta có:
\(\left(\frac{2x}{x+y}\right)^n+\left(\frac{2y}{x+y}\right)^n\ge2\) Hay \(\frac{a^n+b^n}{2}\ge\left(\frac{a+b}{2}\right)^n\)
Cho \(a,b\ge0;\)\(n\in N.\)Chứng minh rằng :
\(\left(\frac{a+b}{2}\right)^n\le\frac{a^n+b^n}{2}.\)
Cho a\(\ge b\),\(b\ge0\),n\(\in N\). Chứng minh rằng :\(\left(\frac{a+b}{2}\right)^n\le\frac{a^n+b^n}{2}\)
cho a,b>0, n \(\in\)N. CMR \(\left(\frac{a+b}{2}\right)^n\le\frac{a^n+b^n}{2}\)
Chứng minh các bất đẳng thức
a, \(y^8-y^7+y^2-y+1>0\)
b, \(m^2+n^2+p^2+q^2+1\ge m\left(n+p+q+1\right)\)
c, \(\left(ax+by\right)^2\le\left(a^2+b^2\right)\left(x^2+y^2\right)\)
d, \(\frac{a^3+b^3}{2}\ge\left(\frac{a+b}{2}\right)^3\)với \(a,b\ge0\)
a) \(a^2+b^2-2ab\ge0\)
b) \(\frac{a^2+b^2}{2}\ge ab\)
c) \(a\left(a+2\right)< \left(a+1\right)^2\)
d) \(m^2+n^2+2\ge2\left(m+n\right)\)
e) \(\left(a+b\right)\left(\frac{1}{a}+\frac{1}{b}\right)\ge4\left(a>0,b>0\right)\)
Giúp mình giải bài này nhé
Let \(a,b,c,k\) be positive real numbers such that \(k\left(ab+bc+ca\right)+2abc\le k^3\) . Prove that:
\(\left(1\right)k\left(a+b+c\right)\ge2\left(ab+bc+ca\right)\)
\(\left(2\right)k\left(a^3+b^3+c^3\right)\ge2\left(a^2b^2+b^2c^2+c^2a^2\right)\)
\(\left(3\right)k\left(a^{2n-1}+b^{2n-1}+c^{2n-1}\right)\ge2\left(a^nb^n+b^nc^n+c^na^n\right)\) \(\left(n\ge0;n\in R\right)\)
Cho a;b;c \(\ge0\).CMR \(\frac{a^3}{b\left(b+c\right)}+\frac{b^3}{c\left(c+a\right)}+\frac{c^3}{a\left(a+b\right)}\)\(\ge\frac{1}{2}\left(a+b+c\right)\)
\(a,5\left(2-3n\right)+42+3n\ge0\)
\(b,\left(n+1\right)^2-\left(2+n\right)\left(2-n\right)\le15\)
\(Cho:a;b\ge0.\)
\(CMR:\frac{a^3+b^3}{2}\ge\left(\frac{a+b}{2}\right)^3\)
