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Mathematics
Mathematics
Problem
Define the function
f
f
f
by
f
(
x
)
=
{
x
2
,
x
<
0
,
3
,
x
=
0
,
∣
x
∣
,
0
<
x
<
1
,
0
,
x
=
1
,
sin
(
π
x
)
,
x
>
1.
f(x)= \begin{cases} x^{2}, & x<0,\\ 3, & x=0,\\ |x|, & 0<x<1,\\ 0, & x=1,\\ \sin(\pi x), & x>1. \end{cases}
f
(
x
)
=
⎩
⎨
⎧
x
2
,
3
,
∣
x
∣
,
0
,
sin
(
π
x
)
,
x
<
0
,
x
=
0
,
0
<
x
<
1
,
x
=
1
,
x
>
1.
Let
g
(
x
)
=
∫
−
2
x
f
(
t
)
d
t
g(x)=\displaystyle\int_{-2}^{x} f(t)\,dt
g
(
x
)
=
∫
−
2
x
f
(
t
)
d
t
. For which of the following values of
x
x
x
must
g
′
(
x
)
g'(x)
g
′
(
x
)
fail to exist? a)
x
=
0
and
x
=
1
x=0 \text{ and } x=1
x
=
0
and
x
=
1
b)
x
=
1
x=1
x
=
1
only c)
x
=
0
x=0
x
=
0
only d)
g
′
(
x
)
g'(x)
g
′
(
x
)
\text{ exists for all real } x
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