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Moldova Contests
Moldova National Olympiad
2006 Moldova National Olympiad
11.2
Exists c, f'(c)=...
Exists c, f'(c)=...
Source: Moldavian MO 2006
March 19, 2006
function
limit
real analysis
real analysis unsolved
Problem Statement
Function
f
:
[
a
,
b
]
→
R
f: [a,b]\to\mathbb{R}
f
:
[
a
,
b
]
→
R
,
0
<
a
<
b
0<a<b
0
<
a
<
b
is continuous on
[
a
,
b
]
[a,b]
[
a
,
b
]
and differentiable on
(
a
,
b
)
(a,b)
(
a
,
b
)
. Prove that there exists
c
∈
(
a
,
b
)
c\in(a,b)
c
∈
(
a
,
b
)
such that
f
′
(
c
)
=
1
a
−
c
+
1
b
−
c
+
1
a
+
b
.
f'(c)=\frac1{a-c}+\frac1{b-c}+\frac1{a+b}.
f
′
(
c
)
=
a
−
c
1
+
b
−
c
1
+
a
+
b
1
.
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