Only the variable with nonzero partial derivative is guaranteed solvable but fixing the other input to zero still produces a valid pair

Let $F : \mathbb{R}^{3} \to \mathbb{R}$ 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 $\varepsilon>0$ , define $\Omega_\varepsilon=(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon)\subset\mathbb{R}^{2}$ . Which of the following statements are necessarily true?

A.There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $g:\Omega_\varepsilon\to(-\delta,\delta)$ such that g(0,0)=0 and F(x,y,g(x,y))=−1 for all $(x,y)\in\Omega_\varepsilon$ .✓
B.There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $h:\Omega_\varepsilon\to(-\delta,\delta)$ such that h(0,0)=0 and F(x,h(x,z),z)=−1 for all $(x,z)\in\Omega_\varepsilon$ .
C.There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $k_{1},k_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $k_{1}(0)=k_{2}(0)=0$ and $F(x,k_{1}(x),k_{2}(x))=-1$ for all $x\in(-\varepsilon,\varepsilon)$ .✓
D.There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $j_{1},j_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $j_{1}(0)=j_{2}(0)=0$ and $F(j_{1}(z),j_{2}(z),z)=-1$ for all $z\in(-\varepsilon,\varepsilon)$ .✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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Last revised 10 Sept 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

Only the variable with nonzero partial derivative is guaranteed solvable but fixing the other input to zero still produces a valid pair

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