A continuous increasing function $f ( x )$ on the closed interval $[ 0,1 ]$ satisfies $$\int _ { 0 } ^ { 1 } f ( x ) d x = 2 , \quad \int _ { 0 } ^ { 1 } | f ( x ) | d x = 2 \sqrt { 2 }$$ When the function $F ( x )$ is defined as $$F ( x ) = \int _ { 0 } ^ { x } | f ( t ) | d t \quad ( 0 \leq x \leq 1 )$$ what is the value of $\int _ { 0 } ^ { 1 } f ( x ) F ( x ) d x$? [4 points] (1) $4 - \sqrt { 2 }$ (2) $2 + \sqrt { 2 }$ (3) $5 - \sqrt { 2 }$ (4) $1 + 2 \sqrt { 2 }$ (5) $2 + 2 \sqrt { 2 }$
A continuous increasing function $f ( x )$ on the closed interval $[ 0,1 ]$ satisfies
$$\int _ { 0 } ^ { 1 } f ( x ) d x = 2 , \quad \int _ { 0 } ^ { 1 } | f ( x ) | d x = 2 \sqrt { 2 }$$
When the function $F ( x )$ is defined as
$$F ( x ) = \int _ { 0 } ^ { x } | f ( t ) | d t \quad ( 0 \leq x \leq 1 )$$
what is the value of $\int _ { 0 } ^ { 1 } f ( x ) F ( x ) d x$? [4 points]\\
(1) $4 - \sqrt { 2 }$\\
(2) $2 + \sqrt { 2 }$\\
(3) $5 - \sqrt { 2 }$\\
(4) $1 + 2 \sqrt { 2 }$\\
(5) $2 + 2 \sqrt { 2 }$