A directional derivative of $f(x,y)=x^2-3y^3$ at the point $(2,1)$ in some direction might be: A directional derivative of 
$$
f(x,y)=x^2-3y^3 
$$
at the point $P(2,1)$ in some direction might be:
a) $-9$
b) $-10$
c) $6$
d) $11$
e) $0$
I'd say it's $-9$ for sure, but what about $0$ (the direction would be $<0,0>$)?
Are there any other proper answers?

$$D_{\vec u}f(\vec x)=\nabla f_{(2,1)}\frac{\vec u}{||\vec u||}\cdot=4u_1-9u_2\;\;\wedge\;\;u_1^2+u_2^2=1$$
so you get a non-linear system of equations
$$\begin{align*}\text{I}&\;\;4u_1-9u_2=t\\\text{II}&\;\;\;\;u_1^2+\;u_2^2=\,1\end{align*}$$
and from here we get
$$u_1^2+\left(\frac{4u_1-t}{9}\right)^2=1\implies 97u_1^2-8tu_1+(t^2-81)=0$$
The above quadratic's discriminant is
$$\Delta=-324(t^2-97)\ge 0\iff |t|\le\sqrt{97}$$
Thus, the system has a solution for any $\;t\in\Bbb R\;\;,\;\;|t|\le\sqrt{97}\;$ , so all of $\,(a), (c), (e)\;$ fulfill this condition .
