Attempt:
a) Let $f(x)=5x-x^2$. Notice that $g_1(x)\cap f(x)=0$. Therefore if there exists a maximum of $f(x)$ it must be $\overline x=0$.
b) I think it refers to the constrait qualifications: Slatters, Zangwill, Kuhn-Tucker, Cotters and l.i..
I don't quite understand what is asked to do, Should I first substitute $\overline x=0$ in each constraint qualification and then to verify that $\exists x\in X:g_1(x)<0,\forall i\in I,\overline D=G',\overline A=G',\overline G_0=G'$ and $\{\nabla g_1(\overline x),i\in I\}$ is l.i., respectively ?
or what should I do?
c) I think the necessary conditions are this
In this case we are in $\mathbb R^2$, hence $\nabla f(\overline x)=(5\ 0)^t,\nabla g_1(\overline x)=(1 \ 0)^t$. When I solve the system to verify the existence of $u\ge 0$, I get that $u=-5$, but this is not correct.
Also I don't know where is this $T=G'$ applied? or how will I applied?
Could someone explain please??
Thank you!