Why does the gradient indicate ascent?

Question

I understand why the gradient gives the greatest directional derivative, as a dot product between the direction and the gradient. But I usually see the gradient called as the direction of steepest ascent. Why does it specifically indicate ascent, as opposed to for instance steepest descent?

Answer 1

Locally, a differentiable function $f:\mathbb{R}^n\to\mathbb{R}$ can be expressed as a Taylor series $$ f(x+hu) = f(x) +h(\nabla f(x)\cdot u) + O(h^2), $$ where $u $ is a unit vector in $\mathbb{R}^n$. For sufficiently small $h$, if one wishes to maximize $f(x+hu)$, then one must maximize $\nabla f(x)\cdot u$. This maximum is acheived exactly at $u = \nabla f(x) / \lvert\lvert \nabla f(x)\rvert\vert$.

If $\nabla f(x)\neq 0$, this direction indicates ascent as opposed to descent because $$ \nabla f(x) \cdot \frac{\nabla f(x)} {\lvert\lvert \nabla f(x)\rvert\vert} = \lvert\lvert\nabla f(x)\rvert\vert > 0, $$ whereas $$ \nabla f(x) \cdot \left(-\frac{\nabla f(x)} {\lvert\lvert \nabla f(x)\rvert\vert}\right) = -\lvert\lvert\nabla f(x)\rvert\vert < 0 $$