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Joint Entrance Examination
Mathematics
Limits, Continuity and Differentiability
Question 376
Question 376 of 401
Medium
MCQ
Suppose \[f: \mathbb{R} \rightarrow(0, \infty)\] be a differentiable function such that \[5 f(x+y)=f(x) \cdot f(y), \forall x, y \in \mathbb{R}\]. If
f
(
3
)
=
320
f(3)=320
f
(
3
)
=
320
, then \[\sum_\limits{n=0}^{5} f(n)\] is equal to :
A
6875
B
6525
C
6575
D
6825
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Official solution
5
f
(
x
+
y
)
=
f
(
x
)
.
f
(
y
)
5f(x + y) = f(x).f(y)
5
f
(
x
+
y
)
=
f
(
x
)
.
f
(
y
)
5
f
(
3
)
=
f
(
1
)
.
f
(
2
)
5f(3) = f(1).f(2)
5
f
(
3
)
=
f
(
1
)
.
f
(
2
)
5
f
(
2
)
=
(
f
(
1
)
)
2
5f(2) = {(f(1))^2}
5
f
(
2
)
=
(
f
(
1
)
)
2
f
(
10
)
=
5
f(10) = 5
f
(
10
)
=
5
f
(
1
)
=
20
f(1) = 20
f
(
1
)
=
20
\[ \Rightarrow f(1).{{{{(f(1))}^2}} \over 5} = 1600\] \[\sum\limits_{n = 0}^5 {f(n) = f(0) + 20 + 80 + 320 + 1280 + 5120} \]
=
1750
+
5120
=
6825
= 1750 + 5120 = 6825
=
1750
+
5120
=
6825
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