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?