Consider a smooth function $F : \Bbb{R}^n \to \Bbb{R}^m$ where $n \geq m$ are positive integers. Consider a curve $\alpha : [0,1 ) \to F^{-1}(0)$ such that $\alpha(0) = x \in F^{-1}(0)$. Then $F(\alpha(t)) = 0$ where $t \in [0,1)$. Differentiating this and looking at the conditions on $\dot \alpha(0)$, $\text{d} F_{\alpha(t)} (\dot \alpha(t)) |_{t=0} =0$ implying that $\dot \alpha(0) \in \text{Ker } \text{d} F_x$. If the first derivative is surjective the Inverse Function Theorem says that $\text{Ker } \text{d} F_x$ defines he whole tangent space at $x$.
The second derivative of $F(\alpha(t)) = 0$ at $t=0$ reads $$ \text{d}^2 F_{x} (\dot \alpha(0),\dot \alpha(0)) + \text{d} F_{x} (\ddot \alpha(0)) = 0 \, . $$ If $h \in \text{Ker } \text{d} F_x$ such that $\text{proj}_{\text{coKer } \text{d}F_x} \text{d}^2 F(h,h) = 0 $ and the function $ \text{proj}_{\text{coKer } \text{d} F_x} \, \text{d}^2 F_{x} (h) = \Bbb{R}^m / \text{Im } \text{d}F_x $ then is $ \text{Ker }( \text{proj}_{\text{coKer } \text{d} F_x} \, \text{d}^2 F_{x} (h) , \text{d}F_x ) $ is a subset of the tangent space at $x$? If not then what conditions on $h$ are required to make $ \text{Ker }( \text{proj}_{\text{coKer } \text{d} F_x} \, \text{d}^2 F_{x} (h) , \text{d}F_x ) $ a subset of the tangent space at $x$?
Any help or guidance solving this problem is highly appreciated.
Note: my derivatives of $F$ are directional derivatives.
It is true that $ \text{Ker }( \text{proj}_{\text{coKer } \text{d} F_x} \, \text{d}^2 F_{x} (h) , \text{d}F_x ) $ is a subset of the tangent space at $x$ given $h \in \text{Ker } \text{d} F_x$ such that $\text{proj}_{\text{coKer } \text{d}F_x} \text{d}^2 F(h,h) = 0 $ and the function $ \text{proj}_{\text{coKer } \text{d} F_x} \, \text{d}^2 F_{x} (h) = \Bbb{R}^m / \text{Im } \text{d}F_x $. This was proven by Avakov in the 1985 paper ''Extremum conditions for smooth problems with equality-type constraints.'' For those who wish to read more look at qualification conditions for derivatives to locally determine the tangent space.