About the solution to question:
For a,b,c>0 and abc=1, prove
a^(b+c) b^(c+a) c^(a+b) ≤1
Way 1:
Let′s say a≤b≤c.
We can prove that a≤1:
If a>1, we will get b≥a>1, c≥b>1,
⇒abc>1
but abc=1!
so a>1 is not true, i.e. a≤1.
Similarly we can also prove that c≥1:
If c<1, we will get b≤c<1, a≤b<1,
⇒abc<1
but abc=1
so c<1 is not true, i.e. c≥1.
We know also
if p≤1, then p^x ≤1 for x≥0
if p≥1, then p^x ≥1 for x≥0
S=a^(b+c) b^(c+a) c^(a+b) =a^(b+c) ((1/(ac)))^(c+a) c^(a+b)
=(a^(b−a) /c^(c−b) )
since a≤1 and b−a≥0, we have
a^(b−a) ≤1
since c≥1 and c−b≥0, we have
c^(b−a) ≥1
⇒S= (a^(b−a) /c^(c−b) )=((≤1)/(≥1))≤1
Way 2:
S=a^(b+c) b^(c+a) c^(a+b) =((a^(a+b+c) b^(c+a+b) c^(a+b+c) )/(a^a b^b c^c ))
=(((abc)^(a+b+c) )/(a^a b^b c^c ))=(1/(a^a b^b c^c ))=(1/(a^a b^b ((1/(ab)))^(1/(ab)) ))
=(((ab)^(1/(ab)) )/(a^a b^b ))
let′s look at function F(x,y)=(((xy)^(1/(xy)) )/(x^x y^y )),
the graph of F(x,y) see comment.
It has a maximum at (1,1) which
is F_(max) =1.
Hence for x, y>0, 0<F(x,y)≤1
⇒S=(((ab)^(1/(ab)) )/(a^a b^b ))=F(a,b)=F(b,a)≤1
|