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Problem
Problem
Let
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
be differentiable functions such that
f
′
(
x
)
g
(
x
)
≠
f
(
x
)
g
′
(
x
)
f'(x)g(x) \neq f(x)g'(x)
f
′
(
x
)
g
(
x
)
=
f
(
x
)
g
′
(
x
)
for any real
x
x
x
. Show that between any two real solutions of
f
(
x
)
=
0
f(x) = 0
f
(
x
)
=
0
, there is atleast one real solution of
g
(
x
)
=
0
g(x) = 0
g
(
x
)
=
0
.
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