## Norm equals to 1 implies postive gradient

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

Proof so far.

I have $< \nabla f(p) , p > = \frac { \partial f }{ \partial p }(p) = \sum ^n _{i=1} p_i \frac { \partial f }{ \partial x_i }(p)$
$=p_1 \frac { \partial f }{ \partial x_1 }(p) + p_2 \frac { \partial f }{ \partial x_2 }(p) + . . . +p_n \frac { \partial f }{ \partial x_n }(p)$

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