Question

cho a,b,c >0 va a+b+c>=6.tim GTNN cua A= 5a+6b+7c+1/a+8/b+27./c

Answer 1

Áp dụng BĐT AM-GM ta có:

\(A=5a+6b+7c+\frac{1}{a}+\frac{8}{b}+\frac{27}{c}\)

\(=4\left(a+b+c\right)+\left(\frac{1}{a}+a\right)+\left(\frac{8}{b}+2b\right)+\left(\frac{27}{c}+3c\right)\)

\(\ge4\cdot6+2\sqrt{\frac{1}{a}\cdot a}+2\sqrt{\frac{8}{b}\cdot2b}+2\sqrt{\frac{27}{c}\cdot3c}\)

\(\ge24+2+2\cdot4+2\cdot9=52\)

Xảy ra khi \(\frac{1}{a}=a;\frac{8}{b}=2b;\frac{27}{c}=3c\Rightarrow a=1;b=2;c=3\)