$f:\Bbb R^n\mapsto\Bbb R$, is -$\nabla f(x)$ the greatest descent direction? Why?
If you consider $-f$ then $\nabla(-f)$ is the greatest ascent direction of $-f$ and since $\nabla(-f)=-\nabla f$ I see more or less why it is true...
Is there more to it?
Have you heard about directional derivatives? it can be shown that the directions of steepest ascent and descent are opposite each other and parallel to the gradient by parametrising the angle of the direction vector and taking the vector's dot-product with the gradient. It also shows that the gradient is perpendicular to the level-contours.
In case you are doing machine learning, there are a lot of helpful replies on this SE question should you need to check them out.
