Question 9: Let f : \mathbb{R}^2 \to \mathbb{R} be a function for whi…

Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a function for which the following is known.

• At every point, the partial derivatives of $f$ exist and are continuous.
• $f(1,-1) = 2$
• The tangent line to the graph of $f$ at $(1,-1)$ in the direction $(1,0)$ contains the point $(2,-1,6)$.
• The tangent line to the graph of $f$ at $(1,-1)$ in the direction $(0,1)$ contains the point $(1,-2,2)$.

Find the directional derivative of $f$ at $(1,-1)$ in the direction $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.