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Mathematics
Mathematics
Problem
Let
f
(
x
)
f(x)
f
(
x
)
is a real valued function defined by
f
(
x
)
=
x
2
+
x
2
∫
−
1
x
t
f
(
t
)
d
t
+
x
3
∫
−
1
x
f
(
t
)
d
t
f(x) = x^2 + x^2 \int_{-1}^{x} t f(t)\, dt + x^3 \int_{-1}^{x} f(t)\, dt
f
(
x
)
=
x
2
+
x
2
∫
−
1
x
t
f
(
t
)
d
t
+
x
3
∫
−
1
x
f
(
t
)
d
t
, then which of the following hold(s) good? a.
∫
−
1
1
t
f
(
t
)
d
t
=
10
11
\int_{-1}^{1} t f(t) dt = \frac{10}{11}
∫
−
1
1
t
f
(
t
)
d
t
=
11
10
b.
f
(
1
)
+
f
(
−
1
)
=
30
11
f(1) + f(-1) = \frac{30}{11}
f
(
1
)
+
f
(
−
1
)
=
11
30
c.
∫
−
1
1
t
f
(
t
)
d
t
>
∫
−
1
1
f
(
t
)
d
t
\int_{-1}^{1} t f(t) dt > \int_{-1}^{1} f(t) dt
∫
−
1
1
t
f
(
t
)
d
t
>
∫
−
1
1
f
(
t
)
d
t
d.
f
(
1
)
−
f
(
−
1
)
=
20
11
f(1) - f(-1) = \frac{20}{11}
f
(
1
)
−
f
(
−
1
)
=
11
20
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