So as the title suggests I have systems:
System 1 (a convex function $f$):
$$\langle \nabla f(x^*), d \rangle < 0$$
System 2 (all $g_i$'s are convex functions):
$$g_i(x^*) + \langle \nabla g_i(x^*), d \rangle < 0 \quad \quad \forall i\in[m]$$
$$g_i(x^*) \leq 0$$
I have to show that no such $d$ exists which will solve both the systems if $x^*$ is a local minimum? I hve tried a lot but could not solve. Any hint would be appreciated. Refere to this video (6:50)