is this sets $A(x) $ ,$B(x)$cone?

Question

$A(x)=\{d\in \mathbb{R^n}: \nabla f(x)*d < 0\}$

$B(x)=\{d\in \mathbb{R^n}: \forall i $ $s.t. g_i(x)=0 , \nabla g_i(x)*d < 0\}$

A,B is a set related below optimization problem

$\min f(x)$

s.t. $g_i(x) <=0$

$x\in\mathbb{R^n}$

$\forall i\in\{1,2,....,p\}$

C is cone when $\forall d\in C $ and $\forall \alpha>=0$ then $\alpha*d\in C$ but in A,B for $,\alpha=0$ don't have $\alpha*d\in C$ , I think it's not cone, is this true?