Cho `a , b , c > 0` . Chứng minh rằng : `P = (a(b + c))/(a^2 + (b + c)^2) + (b(a + c))/(b^2 + (a + c)^2) + (c(a + b))/(c^2 + (a + b)^2) ≤ 6/5`

0 bình luận về “Cho `a , b , c > 0` . Chứng minh rằng : `P = (a(b + c))/(a^2 + (b + c)^2) + (b(a + c))/(b^2 + (a + c)^2) + (c(a + b))/(c^2 + (a + b)^2) ≤ 6/5`”

1. `a^2+(b+c)^2=a^2+((b+c)^2)/4 +(3(b+c)^2)/4≥2(a(b+c))/2 +(3(b+c)^2)/4=(4a(b+c))/4+(3(b+c)^2)/4=(b+c)/4 (4a+3(b+c))`

   `⇔1/(a^2+(b+c)^2)≤4/((b+c)(4a+3(b+c)))`

   `⇔(a(b+c)/(a^2+(b+c)^2)≤(4a)/(4a+3b+3c)≤1/(25) ((9^2a)/(3a+3b+3c))+a/a)=(27a)/(25a+25b+25c)+1/(25)`

   CM tương tự 

   `⇒(b(a+c)/(b^2+(a+c)^2)≤(27b)/(25a+25b+25c)+1/(25)`

   `⇒(c(b+a)/(c^2+(b+a)^2)≤(27c)/(25a+25b+25c)+1/(25)`

   `⇒P≤(27a)/(25a+25b+25c)+1/(25)+(27b)/(25a+25b+25c)+1/(25)+(27c)/(25a+25b+25c)+1/(25)`

   `⇒P≤(27)/(25)+3/(25)=(30)/(25)=6/5`

   `”=”`xẩy ra khi :

   `a=b=c`

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