Let $f , g : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R }$ be functions. Let $F = ( f , g ) : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 }$. Assume that $F$ is infinitely differentiable and that $F ( 0,0 ) = ( 0,0 )$. Suppose further that the function $f g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$ is everywhere non-negative. Then (A) $f _ { x } ( 0,0 ) = 0 , f _ { y } ( 0,0 ) = 0$. (B) $g _ { x } ( 0,0 ) = 0 , g _ { y } ( 0,0 ) = 0$. (C) The image of $F$ is not dense in $\mathbb { R } ^ { 2 }$. (D) $\operatorname { det } J ( 0,0 ) = 0$ where $J$ is the matrix of first partial derivatives (i.e., the jacobian matrix).
Let $f , g : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R }$ be functions. Let $F = ( f , g ) : \mathbb { R } ^ { 2 } \longrightarrow \mathbb { R } ^ { 2 }$. Assume that $F$ is infinitely differentiable and that $F ( 0,0 ) = ( 0,0 )$. Suppose further that the function $f g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$ is everywhere non-negative. Then\\
(A) $f _ { x } ( 0,0 ) = 0 , f _ { y } ( 0,0 ) = 0$.\\
(B) $g _ { x } ( 0,0 ) = 0 , g _ { y } ( 0,0 ) = 0$.\\
(C) The image of $F$ is not dense in $\mathbb { R } ^ { 2 }$.\\
(D) $\operatorname { det } J ( 0,0 ) = 0$ where $J$ is the matrix of first partial derivatives (i.e., the jacobian matrix).