Problem

Find the critical points of the function

$$f(x,y)=x^3-y^2-xy+1.$$

(a) $(0,0)$

(b) $(0,0)$ and $\left(-\frac16,\frac1{12}\right)$

(c) $(0,0)$ and $\left(\frac16,-\frac1{12}\right)$

(d) Does not exist.

(e) None of the above.