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3
Monotony of a rational cyclic function
Monotony of a rational cyclic function
Source: Romanian NO 2011, grade x, p. 3
October 3, 2019
function
algebra
cyclic function
Problem Statement
Let be three positive real numbers
a
,
b
,
c
.
a,b,c.
a
,
b
,
c
.
Show that the function
f
:
R
⟶
R
,
f:\mathbb{R}\longrightarrow\mathbb{R} ,
f
:
R
⟶
R
,
f
(
x
)
=
a
x
b
x
+
c
x
+
b
x
a
x
+
c
x
+
c
x
a
x
+
b
x
,
f(x)=\frac{a^x}{b^x+c^x} +\frac{b^x}{a^x+c^x} +\frac{c^x}{a^x+b^x} ,
f
(
x
)
=
b
x
+
c
x
a
x
+
a
x
+
c
x
b
x
+
a
x
+
b
x
c
x
,
is nondecresing on the interval
[
0
,
∞
)
\left[ 0,\infty \right)
[
0
,
∞
)
and nonincreasing on the interval
(
−
∞
,
0
]
.
\left( -\infty ,0 \right] .
(
−
∞
,
0
]
.
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