Show That F Is Conservative and Find a Function F $\mathbf { F } = \nabla f$

Multiple Choice

Show that F is conservative and find a function f such that $\mathbf { F } = \nabla f$ , and use this result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where C is any path from $A \left( x _ { 0 } , y _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } \right)$ . $\mathbf { F } ( x , y ) = \left( 12 x ^ { 2 } y ^ { 2 } - 16 x y ^ { 3 } \right) \mathbf { i } + \left( 8 x ^ { 3 } y - 24 x ^ { 2 } y ^ { 2 } \right) \mathbf { j }$ ; $A ( 3,1 )$ and $B ( 3,2 )$

A) $\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 414$
B) $\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 152$
C) $\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 236$
D) $\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 180$