Cho biểu thức:
A= $\frac{a}{a+b+c}$ + $\frac{b}{b+c+d}$ + $\frac{c}{c+d+a}$ + $\frac{d}{d+a+b}$
(a; b; c; d khác 0)
CMR: 1
Cho biểu thức:
A= $\frac{a}{a+b+c}$ + $\frac{b}{b+c+d}$ + $\frac{c}{c+d+a}$ + $\frac{d}{d+a+b}$
(a; b; c; d khác 0)
CMR: 1
Ta có:
$\dfrac{a}{a+b+c+d} < \dfrac{a}{a+b+c} < \dfrac{a}{a + c}$
$\dfrac{b}{a+b + c + d} < \dfrac{b}{b+c+d} < \dfrac{b}{b+d}$
$\dfrac{c}{a+b+c+d} <\dfrac{c}{c + d + a}<\dfrac{c}{a+c}$
$\dfrac{d}{a+b+c + d}<\dfrac{d}{d+a+b} <\dfrac{d}{b + d}$
Cộng vế theo vế ta được:
$\dfrac{a}{a+b+c+d}+ \dfrac{b}{a+b+c+d} + \dfrac{c}{a+b+c+d} + \dfrac{d}{a+b+c+d} < A <\dfrac{a}{a+c} + \dfrac{b}{b + d} +\dfrac{c}{a + c} + \dfrac{d}{b+d}$
$\Leftrightarrow 1 < A < 2\quad (đpcm)$
Đáp án :
`1<A<2`
Các bước giải tương ứng :
`+)`Ta có :
`a/(a+b+c+d)<a/(a+b+c)<a/(a+b)`
`b/(a+b+c+d)<b/(b+c+d)<b/(a+b)`
`c/(a+b+c+d)<c/(c+d+a)<c/(c+d)`
`d/(a+b+c+d)<d/(d+a+b)<d/(c+d)`
`=>a/(a+b+c+d)+b/(a+b+c+d)+c/(a+b+c+d)+d/(a+b+c+d)<a/(a+b+c)+b/(b+c+d)+c/(c+d+a)+d/(d+a+b)<a/(a+b)+b/(a+b)+c/(c+d)+d/(c+d)`
`=>(a+b+c+d)/(a+b+c+d)<A<(a+b)/(a+b)+(c+d)/(c+d)`
`=>1<A<1+1`
`=>1<A<2`
Vậy : `1<A<2`