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Problem
Problem
If
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
are continuous functions in
[
a
,
b
]
[a, b]
[
a
,
b
]
and they are differentiable in
(
a
,
b
)
(a, b)
(
a
,
b
)
, then prove that there exists
c
∈
(
a
,
b
)
c \in (a, b)
c
∈
(
a
,
b
)
such that
∣
f
(
a
)
f
(
b
)
g
(
a
)
g
(
b
)
∣
=
(
b
−
a
)
∣
f
(
a
)
f
′
(
c
)
g
(
a
)
g
′
(
c
)
∣
\left| \begin{array}{cc} f(a) & f(b) \\ g(a) & g(b) \end{array} \right| = (b-a) \left| \begin{array}{cc} f(a) & f'(c) \\ g(a) & g'(c) \end{array} \right|
f
(
a
)
g
(
a
)
f
(
b
)
g
(
b
)
=
(
b
−
a
)
f
(
a
)
g
(
a
)
f
′
(
c
)
g
′
(
c
)
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