A problem related to multivariate differential calculus. I am doubted with this following problem and can not understand how to proceed. Please help a little for solving the problem and the counterexample with it. I can not construct. 
Prove that there is no real valued function $f$ such that $f'(c, u) > 0$ for a fixed point $c$ in $\mathbb{R}^n$ and every non-zero vector $u$ in $\mathbb{R}^n$. 
Give an example such that $f'(c, u) > 0 $ for a fixed direction $u$ and for every $c$ in $\mathbb{R}^n$.
Thank you. 

The derivative operator $df_c : T_c\mathbb{R}^n \to T_{f(c)}\mathbb{R}$ is linear.  If it takes on a "positive" value at $u$ (we have to identify the tangent space to the reals at a point, which is a vector space, with the reals themselves, which have a notion of positive and negative), then it takes on the negative of that value at $-u$.
Understanding what it means for the derivative at a point to be a linear map is probably the most important thing you need to know to move forward right now, so go ahead an spend some time thinking about the meaning and implications.
