2.1 Identities and Inequalities 29Example.Find the minimum of the functionf:( 0 ,∞)^3 →R,
f (x, y, z)=xz+yz−(xy)z/^4.Solution.Rewrite the function as
f (x, y, z)=(xz/^2 −yz/^2 )^2 + 2[
(xy)z/^4 −1
4
] 2
−
1
8
.
We now see that the minimum is−^18 , achieved if and only if(x,y,z)=(a, a,loga 161 ),
wherea∈( 0 , 1 )∪( 1 ,∞).
We continue with a problem from the 2001 USA team selection test proposed also by
the second author of the book.
Example.Let(an)n≥ 0 be a sequence of real numbers such that
an+ 1 ≥a^2 n+1
5
, for alln≥ 0.Prove that
√
an+ 5 ≥a^2 n− 5 , for alln≥5.Solution.It suffices to prove thatan+ 5 ≥an^2 , for alln≥0. Let us write the inequality
for a number of consecutive indices:
an+ 1 ≥an^2 +1
5
,
an+ 2 ≥an^2 + 1 +1
5
,
an+ 3 ≥an^2 + 2 +1
5
,
an+ 4 ≥an^2 + 3 +1
5
,
an+ 5 ≥an^2 + 4 +1
5
.
If we add these up, we obtain
an+ 5 −an^2 ≥(a^2 n+ 1 +an^2 + 2 +an^2 + 3 +an^2 + 4 )−(an+ 1 +an+ 2 +an+ 3 +an+ 4 )+ 5 ·1
5
=
(
an+ 1 −1
2
) 2
+
(
an+ 2 −1
2
) 2
+
(
an+ 3 −1
2
) 2
+
(
an+ 4 −1
2
) 2
≥ 0.
The conclusion follows.