Let $A $ be an invertible real $ n\times n $ matrix. Define a function $ F\colon\mathbb R^{n}\times \mathbb R^{n} \to\mathbb R $ by $ F (x,y)=\langle Ax,y\rangle $ where $ \langle x,y\rangle $ denotes the inner product of $x$ and $y$ .let $ DF(x,y) $ denote the derivative of $F$ at (x,y) which is a linear transformation from $\mathbb R^{n}\times \mathbb R^{n}\to \mathbb R $ then
If $ x\neq 0 $ then $ DF(x,0)\neq 0 $
If $ y\neq 0 $ then $ DF(0,y)\neq 0 $
If $ (x,y)\neq 0 $ then $ DF(x,y)\neq 0 $
If $ x=0 $ or $ y=0 $ then $ DF(x,y) $=0 $
To determine which options are true , I use the fact $ (DF (x,y))(h,k) = f(h,y)+f(x,k) = \langle Ah,y\rangle +\langle Ax,k\rangle $ . So option 4 is true , and the other 3 options are not necessarily true, since it depends on $ (h,k) $ also.
Am I right ? otherwise please give me some hints. Thanks.