An exercise problem states, Which of the systems of nonlinear equations can be expressed as $∇φ(x) = 0$ for some function φ to be minimized?
What does $∇φ(x) = 0$ mean? Here are the nonlinear equations they're talking about (not asking you to solve it for me, just want to understand what the question is asking me to do)
Imagine a function $\phi(x_1, x_2)$, what you want to check if whether it is possible to find such a function such that when you take the derivative with respect to $x_1$ you get one of the equations, and when you take it with respect to $x_2$ you get the other one. Here is an example for (a)
$$ \frac{\partial \phi}{\partial x_1} = x_1^2 + x_1x_2^3 - 9 = 0 $$
this implies that
$$ \phi(x_1,x_2) = \frac{1}{3}x_1^3 + \frac{1}{2}x_1^2 x_2^3 - 9 x_1+ f(x_2) $$
then, taking the derivative with respect to $x_2$ we get
$$ \frac{\partial \phi}{\partial x_2} = \frac{3}{2}x_1^2x_2^2 + f'(x_2) = 0 \tag{1} $$
while the second equation is
$$ 3x_1^2x_2 - x_2^3 - 4 = 0 \tag{2} $$
You see that it is impossible to find a function of $x_2$ that transform Eq (1) into Eq (2). You can also argue that the first equation is actually the result of taking $\partial \phi/\partial x_2$. I will honor your request and will not complete the analysis for you