Question

Determine whether the vector field is conservative and if so find the potential function, f(x,y)=(6x 5y)i

Answer 1

I'm assuming you mean

$\vec f(x,y) = (6x + 5y)\,vec\imath$

It's a bit odd that there is no $\vec\jmath$-component, but at any rate we want to find a scalar function $f(x,y)$ for which

$\nabla f(x,y) = \vec f(x,y)$

This would mean

$\dfrac{\partial f}{\partial x} = 6x + 5y$

and

$\dfrac{\partial f}{\partial y} = 0$

The second equation tells us $f(x,y)$ is a function that only depends on $x$. But the first equation tells us $f(x,y)$ is a function of both $x$ and $y$. No such function exists, so the given field is not conservative.