Locate and Classify All Critical Points for the Given Function $f(x, y)=x^{2}+x+y^{2}-y-5$

Multiple Choice

Locate and classify all critical points for the given function.
- $f(x, y) =x^{2}+x+y^{2}-y-5$

A) Minimum of $-\frac{7}{2}$ at $\left(-\frac{1}{2},-\frac{1}{2}\right)$
B) Maximum of $-\frac{7}{2}$ at $\left(-\frac{1}{2},-\frac{1}{2}\right)$
C) Minimum of $-\frac{7}{2}$ at $\left(-\frac{1}{2}, \frac{1}{2}\right)$
D) Maximum of $-\frac{7}{2}$ at $\left(-\frac{1}{2}, \frac{1}{2}\right)$