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Mathematics
Problem
Let
f
(
t
)
=
t
2
f(t) = t^2
f
(
t
)
=
t
2
for
0
≤
t
≤
1
0 \leq t \leq 1
0
≤
t
≤
1
, and
g
(
t
)
=
t
3
g(t) = t^3
g
(
t
)
=
t
3
for
0
≤
t
≤
1
0 \leq t \leq 1
0
≤
t
≤
1
. The value of
c
c
c
with
0
<
c
<
1
0 < c < 1
0
<
c
<
1
at which
f
(
1
)
−
f
(
0
)
g
(
1
)
−
g
(
0
)
=
f
′
(
c
)
g
′
(
c
)
\frac{f(1) - f(0)}{g(1) - g(0)} = \frac{f'(c)}{g'(c)}
g
(
1
)
−
g
(
0
)
f
(
1
)
−
f
(
0
)
=
g
′
(
c
)
f
′
(
c
)
is (a)
2
3
\frac{2}{3}
3
2
(b)
1
3
\frac{1}{3}
3
1
(c)
1
2
\frac{1}{2}
2
1
(d)
1
6
\frac{1}{6}
6
1
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