Suppose that $\displaystyle f: \mathbb {R} ^n \rightarrow \mathbb {R} $ is continuously differentiable, and let $\displaystyle p \in \mathbb {R} ^n $ such that $\displaystyle ||p|| = 1$. Prove that $\displaystyle < \nabla f(p) , p > > 0 $

Proof so far.

I have $\displaystyle < \nabla f(p) , p > = \frac { \partial f }{ \partial p }(p) = \sum ^n _{i=1} p_i \frac { \partial f }{ \partial x_i }(p) $

$\displaystyle =p_1 \frac { \partial f }{ \partial x_1 }(p) + p_2 \frac { \partial f }{ \partial x_2 }(p) + . . . +p_n \frac { \partial f }{ \partial x_n }(p) $

So the sum of the $\displaystyle p_i$ is positive, but are the partial derivatives positive as well?