Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. (If the vector field is not conservative, enter DNE.) F(x, y) = (3x2 − 2y2)i + (4xy + 4)j

In this problem, since the condition of equal derivatives does not apply, the vector field is not conservative.

A vector field can be described as:

It is conservative if:

In this problem, the field is:

Then:

Since , the field is not conservative.

A similar problem is given at link

In order for F to be conservative, there must be a scalar function f such that the gradient of f is equal to F. This means

Integrate both sides of the first equation with respect to x :

Differentiate both sides with respect to y :

But we assume g is a function of y, which means its derivative can't possibly contain x, so there is no scalar function f whose gradient is F. Therefore F is not conservative.

what i would say the best answer is a. 40.5