Can a bivariate polynomial be increasing in one non-zero direction only? Consider a bivariate polynomial 
$$f(x,y)=\sum_{i=0}^n \sum_{j=0}^m b_{ij}x^iy^j$$
defined on $[0,1]^2$. Suppose that on $[0,1]^2$ this polynomial weakly increases in direction $d=(d_1,d_2) \neq 0$: 
$$d_1 \frac{\partial f(x,y)}{\partial x} + d_2 \frac{\partial f(x,y)}{\partial y} \geq 0.$$ 
My question is whether it is possible to have a situation when such $d$ is the only non-trivial direction of monotonicity  of $f$? That is, if there is another $\tilde{d} \neq 0$ such that $f$ is increasing in direction $\tilde{d}$  on $[0,1]^2$ then it has to be that 
$$\tilde{d}=\alpha d $$
for some $\alpha>0$? 

I think I figured out an asnwer to my question. Yes, it can happen. E.g., it seem that the polynomial
$$f(x,y)=x y(1-y), \quad (x,y) \in [0,1]^2$$
is increasing in the direction $d=(1,0)$ only. 
