Answer
The answers are (b) and (c).
Work Step by Step
To avoid a change in temperature, the rate of change must be zero. According to Theorem 4, the rate of change of the temperature at $P$ is ${D_{\bf{u}}}f\left( P \right) = ||\nabla {f_P}||\cos \theta $, where ${\bf{u}}$ is unit vector in northeast (NE) direction.
Since $||\nabla {f_P}|| \ne 0$, we require that $\cos \theta = 0$ so that ${D_{\bf{u}}}f\left( P \right) = 0$. Thus, the solutions are $\theta = \pm \frac{\pi }{2}$.
Hence, we should walk in NW (northwest) and SE (southeast) directions to avoid a change in temperature. So, the answers are (b) and (c).
