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Mathematics
Mathematics
Problem
A function
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
satisfies the equation
f
(
x
+
y
)
=
f
(
x
)
⋅
f
(
y
)
f(x+y) = f(x) \cdot f(y)
f
(
x
+
y
)
=
f
(
x
)
⋅
f
(
y
)
for all
x
,
y
x, y
x
,
y
in
R
\mathbb{R}
R
and
f
(
x
)
≠
0
f(x) \neq 0
f
(
x
)
=
0
for any
x
x
x
in
R
\mathbb{R}
R
. Let the function be differentiable at
x
=
0
x=0
x
=
0
and
f
′
(
0
)
=
2
f'(0) = 2
f
′
(
0
)
=
2
. a. Show that
f
′
(
x
)
=
2
f
(
x
)
f'(x) = 2f(x)
f
′
(
x
)
=
2
f
(
x
)
for all
x
x
x
in
R
R
R
. b. Hence, determine
f
(
x
)
f(x)
f
(
x
)
.
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