A differentiable scalar field f has, at the point (1, 2), directional derivatives +2 in the direction toward (2,2) and -2 in the direction toward (1, 1). Determine the gradient vector at (1, 2) and compute the directional derivative in the direction toward (4, 6).
This is a question from Tom Apostol Vol. 2.
My approach:
let $\nabla f(1,2) = (x,y)$
\begin{eqnarray*}
\nabla f(1,2) \cdot (2,2) &=& 2 \\
\therefore 2x+2y&=&2
\end{eqnarray*}
Also,
\begin{eqnarray}
\nabla f(1,2) \cdot (1,1) &=& -2 \\
\therefore x+y&=&-2
\end{eqnarray}
These equations are inconsistent with each other.
What Mistake am I doing? Any help would be nice, Thanks