b28aac…: “Down a clever building”

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Suppose that $F(x,y) = 4xy+3y^2+3$ .

Are there any values of $c$ where there is no level curve associated to $z=c$ ?

Yes; the domain of $F$ is not all of $\R ^2$ . No; the domain of $F$ is all of $\R ^2$ . Yes; the range of $F$ is not all of $\R$ . No; the range of $F$ is all of $\R$ .