\(\frac{a^4}{a^3+b^3} = \frac{a^4 + a^3b^3 - a^3b^3}{a^3+b^3} = a - \frac{a^3b^3}{a^3+b^3}\)

Ta có: \(a^3+b^3\ge ab\left(a+b\right)\)

=>\(\frac{ab^3}{a^3+b^3} \le \frac{ab^3}{ab(a+b)} = \frac{b^2}{a+b}\)

=>\(\frac{a^4}{a^3+b^3} = a - \frac{ab^3}{a^3+b^3} \ge a - \frac{b^2}{a+b}\)

Chứng minh tương tự, ta cũng có:

\(\frac{b^4}{b^3+c^3}\ge b-\frac{c^2}{b+c};\frac{c^4}{c^3+a^3}\ge c-\frac{a^2}{c+a}\)

Do đó: \(\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge\left(a+b+c\right)-\left(\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\right)\)

Theo BĐT Cauchy-Schwart, ta có:

\(\frac{a^2}{b+c} + \frac{b^2}{c+a} + \frac{c^2}{a+b} \ge \frac{(a+b+c)^2}{(b+c) + (c+a) + (a+b)} = \frac{(a+b+c)^2}{2(a+b+c)} = \frac{a+b+c}{2}\)

Do đó: \(\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge\left(a+b+c\right)-\frac{\left(a+b+c\right)}{2}=\frac{a+b+c}{2}\)

Dấu '=' xảy ra khi a=b=c>0

-Áp dụng BĐT Caushy Schwarz ta có:

\(\dfrac{1^2}{a+1}+\dfrac{1^2}{b+1}\ge\dfrac{\left(1+1\right)^2}{a+b+1+1}=\dfrac{4}{3}\)

-Dấu "=" xảy ra khi \(a=b=\dfrac{1}{2}\)

 Nguyễn Việt Lâm  Thầy giúp em được không ạ 

\(\dfrac{a^5}{b^3}+\dfrac{a^5}{b^3}+\dfrac{a^5}{b^3}+\dfrac{a^5}{b^3}+b^2\ge5\sqrt[5]{\dfrac{a^{20}b^2}{b^{12}}}=5.\dfrac{a^4}{b^2}\)

\(\Rightarrow4.\dfrac{a^5}{b^3}+b^2\ge5.\dfrac{a^4}{b^2}\)

Tương tự: \(4.\dfrac{b^5}{c^3}+c^2\ge5\dfrac{b^4}{c^2};4\dfrac{c^5}{a^3}+a^2\ge5.\dfrac{c^4}{a^2}\)

\(\Rightarrow4\left(\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}\right)+a^2+b^2+c^2\ge5\left(\dfrac{c^4}{a^2}+\dfrac{a^4}{b^2}+\dfrac{b^4}{c^2}\right)\)

Lại có: \(\dfrac{a^5}{b^3}+\dfrac{a^5}{b^3}+b^2+b^2+b^2\ge5a^2\)

\(\Rightarrow2.\dfrac{a^5}{b^3}+3b^2\ge5a^2\), tương tự: \(2.\dfrac{b^5}{c^3}+3c^2\ge5b^2;2\dfrac{c^5}{a^3}+3a^2\ge5c^2\)

\(\Rightarrow\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}\ge a^2+b^2+c^2\)

\(\Rightarrow\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}+4.\left(\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}\right)\ge4.\left(\dfrac{a^5}{b^3}+\dfrac{b^5}{c^3}+\dfrac{c^5}{a^3}\right)+a^2+b^2+c^2\ge5.\left(\dfrac{c^4}{a^2}+\dfrac{a^4}{b^2}+\dfrac{b^4}{c^2}\right)\)

\(\Rightarrow dpcm\)

giả sử \(a>b>c>0\) thì ta có :

\(\dfrac{a^4}{b^2}\left(\dfrac{a}{b}-1\right)+\dfrac{b^4}{c^2}\left(\dfrac{b}{c}-1\right)+\dfrac{c^4}{a^2}\left(\dfrac{c}{a}-1\right)\ge\dfrac{2a^2b}{c}+\dfrac{c^5}{a^3}-\dfrac{c^4}{a^2}\)

\(\ge\dfrac{2c^4b}{a}-\dfrac{c^4}{a^2}=\dfrac{c^4}{a}\left(2b-\dfrac{1}{a}\right)>0\)

làm tương tự cho trường hợp \(c>b>a>0\) ; \(b>a>c\) và \(b>c>a\)

\(\Rightarrow\left(đpcm\right)\)

mấy câu cậu câu đăng khác bn làm tương tự nha . nếu bn lm không được thì có j mk lm luôn cho còn h mk bạn rồi :(

day la cac tinh chat ma

a/

\(\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a^3}{b^3}=\frac{c^3}{d^3}\)

Áp dụng tỉ lệ thức ta có:

\(\frac{a^3}{b^3}=\frac{c^3}{d^3}\Rightarrow\frac{a^3}{c^3}=\frac{b^3}{d^3}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\frac{a^3}{c^3}=\frac{b^3}{d^3}=\frac{a^3+b^3}{c^3+d^3}=\frac{a^3-b^3}{c^3-d^3}\)

Vậy \(\frac{a^3+b^3}{c^3+d^3}=\frac{a^3-b^3}{c^3-d^3}\)

Áp dụng  \(x^2+y^2+z^2\ge xy+yz+zx\) và \(x^2+y^2+z^2\ge\dfrac{1}{3}\left(x+y+z\right)^2\)

\(N\ge\dfrac{a^2b}{c}+\dfrac{b^2c}{a}+\dfrac{c^2a}{b}\ge\dfrac{1}{3}\left(a\sqrt{\dfrac{b}{c}}+b\sqrt{\dfrac{c}{a}}+c\sqrt{\dfrac{a}{b}}\right)^2=3\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c=1\)