Show That $\mathrm { F }$ Is Conservative, and Find a Function

Multiple Choice

Show that $\mathrm { F }$ is conservative, and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ .
$\mathbf { F } ( x , y , z ) = 8 x y \mathbf { i } + \left( 4 x ^ { 2 } + 16 y ^ { 3 } z ^ { 2 } \right) \mathbf { j } + 8 y ^ { 4 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( 2 , - 2,2 )$
Select the correct answer.

A) 224
B) 135
C) 117
D) 70