a) \(BĐT\Leftrightarrow\sqrt{c\left(a-c\right)}+\sqrt{c\left(b-c\right)}\le\sqrt{ab}\)

\(\Leftrightarrow\sqrt{\frac{c\left(a-c\right)}{ab}}+\sqrt{\frac{c\left(b-c\right)}{ab}}\le1\)

\(\Leftrightarrow\sqrt{\frac{c}{b}\left(1-\frac{c}{a}\right)}+\sqrt{\frac{c}{a}\left(1-\frac{c}{b}\right)}\le1\)

Áp dụng AM-GM:\(VT\le\frac{1}{2}\left(\frac{c}{b}+1-\frac{c}{a}+\frac{c}{a}+1-\frac{c}{b}\right)=1\left(đpcm\right)\)

Dấu = xảy ra khi (a+b).c=ab

b) \(2+b+c+2+b+c\ge2\sqrt{\left(b+1\right)\left(c+1\right)}+2+b+c=\left(\sqrt{1+b}+\sqrt{1+c}\right)^2\ge4\left(1+a\right)\)

\(\Leftrightarrow b+c\ge2a\)

cau a) dung cosi

\(\sqrt{c\left(a-c\right)}\le\frac{a-c+c}{2}\) ap dung cosi cho hai so c va a-c

tuong tu voi cac so khac

\(BT\le\frac{a-c+c}{2}+\frac{b-c+c}{2}-\frac{a+b}{2}\)(bt la VT cua de)

=> DPCM

b)

dung cosi nhu cau a

lam nhanh luon

\(\sqrt{1+b}\ge\frac{b+1+1}{2}\)

tuong tu

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

<=> b+c>=2a

đánh dấu *

\(\sqrt{1+b}+\sqrt{1+c}=2\sqrt{1+a}\) (1)

⇒ \(b+c+2+2\sqrt{\left(1+b\right)\left(1+c\right)}=4a+4\)

⇒ \(b+c=4a+2-2\sqrt{\left(1+b\right)\left(1+c\right)}\)

Từ (1) ta lại có: \(2\sqrt{1+a}=\sqrt{1+b}+\sqrt{1+c}\ge2\sqrt[4]{\left(1+b\right)\left(1+c\right)}\)

⇒ \(1+a\ge\sqrt{\left(1+b\right)\left(1+c\right)}\) ⇒

\(b+c=4a+2-2\sqrt{\left(1+b\right)\left(1+c\right)}\ge4a+2-2\left(1+a\right)=2a\)

Vậy \(b+c\ge2a\), "=" xảy ra khi \(b=c=a\)

Áp dụng bđt bunhiacopxki ta có:

\(\left(\sqrt{b+1}+\sqrt{c+1}\right)^2\le\left(b+1+c+1\right)\left(1^2+1^2\right)\)

\(\Leftrightarrow2\left(b+c+2\right)\ge4\left(a+1\right)\)

\(\Leftrightarrow b+c+2\ge2a+2\)

\(\Leftrightarrow b+c\ge2a\)