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Mathematics
Mathematics
Problem
Let
F
F
F
be a function with
F
′
(
x
)
=
∣
x
−
1
∣
F'(x)=|x-1|
F
′
(
x
)
=
∣
x
−
1∣
and
F
(
0
)
=
0
F(0)=0
F
(
0
)
=
0
. Which of the following gives
F
(
x
)
F(x)
F
(
x
)
for all real
x
x
x
? a)
F
(
x
)
=
{
x
−
x
2
2
,
x
≤
1
,
(
x
−
1
)
2
2
,
x
≥
1.
\displaystyle F(x)=\begin{cases}x-\dfrac{x^{2}}{2}, & x\le 1,\\ \dfrac{(x-1)^{2}}{2}, & x\ge 1.\end{cases}
F
(
x
)
=
⎩
⎨
⎧
x
−
2
x
2
,
2
(
x
−
1
)
2
,
x
≤
1
,
x
≥
1.
b)
F
(
x
)
=
{
x
−
x
2
2
,
x
≤
1
,
(
x
−
1
)
2
2
+
1
2
,
x
≥
1.
\displaystyle F(x)=\begin{cases}x-\dfrac{x^{2}}{2}, & x\le 1,\\ \dfrac{(x-1)^{2}}{2}+\dfrac12, & x\ge 1.\end{cases}
F
(
x
)
=
⎩
⎨
⎧
x
−
2
x
2
,
2
(
x
−
1
)
2
+
2
1
,
x
≤
1
,
x
≥
1.
c)
F
(
x
)
=
(
x
−
1
)
2
2
−
1
2
\displaystyle F(x)=\dfrac{(x-1)^{2}}{2}-\dfrac12
F
(
x
)
=
2
(
x
−
1
)
2
−
2
1
d)
F
(
x
)
=
{
x
2
2
−
x
,
x
≤
1
,
(
x
−
1
)
2
2
,
x
≥
1.
\displaystyle F(x)=\begin{cases}\dfrac{x^{2}}{2}-x, & x\le 1,\\ \dfrac{(x-1)^{2}}{2}, & x\ge 1.\end{cases}
F
(
x
)
=
⎩
⎨
⎧
2
x
2
−
x
,
2
(
x
−
1
)
2
,
x
≤
1
,
x
≥
1.
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