Let F:R^3toR be a continuously differentiable function such that F(0,0,0)=-1, partial F/partial x(0,0,0)=2, partial F/partial y(0,0,0)=0, and partial F/partial z(0,0,0)=3. For epsilon0, define Omegaepsilon=(-epsilon,epsilon)times(-epsilon,epsilon)subsetR^2. Which of the following statements are necessarily true?

Q69Section C (Part C)69 of 120

Implicit function theorem — which variable can be solved for

Let $F:\mathbb{R}^3\to\mathbb{R}$ be a continuously differentiable function such that $F(0,0,0)=-1$ , $\frac{\partial F}{\partial x}(0,0,0)=2$ , $\frac{\partial F}{\partial y}(0,0,0)=0$ , and $\frac{\partial F}{\partial z}(0,0,0)=3$ . For $\epsilon>0$ , define $\Omega_{\epsilon}=(-\epsilon,\epsilon)\times(-\epsilon,\epsilon)\subset\mathbb{R}^2$ . Which of the following statements are necessarily true?

Multiple select: one or more options are correct. Select all of them, then submit — marks need the exact set.

Select every correct option, then submit.

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