I am solving past calculus exams, and I came across the following question.
Does the equation: $ F(x,y,z) = 2\sin(x^2yz) - 3x + 5y^2 - 2e^{yz} = 0 $ define a differentiable function $z = f(x,y)$ in a neighborhood of $p = (1, 1, 0)$?
At first, I thought this was a natural candidate for the implicit function theorem, but: $ \left.\begin{matrix} \frac{\partial }{\partial z}F \end{matrix}\right|_{(1,1,0)} = \begin{matrix} 2x^2y\cos(x^2yz) -2ye^{yz} \end{matrix}_{(1,1,0)} = 0 $ hence, the theorem doesn't hold in this case.
Ideas? Thanks!