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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}$ be differentiable at $a \\in U$, and suppose that $\\nabla f(a) \\ne 0$.\nThen for each unit vector $u \\in \\mathbb{R}^n$, the [[directional derivative|0Y7PVW4Q44]] satisfies $-\\|\\nabla f(a)\\| \\le D_u f(a) \\le \\|\\nabla f(a)\\|$.\nThe upper bound is attained if and only if $u = \\nabla f(a) / \\|\\nabla f(a)\\|$, and the lower bound if and only if $u = -\\nabla f(a) / \\|\\nabla f(a)\\|$.\n\n## Remarks\n\nThe proof combines $D_u f(a) = \\nabla f(a) \\cdot u$, from the statement that [[the derivative gives the directional derivatives|0A6PMR20N4]], with the Cauchy–Schwarz inequality $|w \\cdot u| \\le \\|w\\| \\, \\|u\\|$ and its case of equality.\nSo the [[gradient|0NS2QYVC4Y]] gives the direction of fastest increase, and its norm is the fastest rate of increase per unit of distance.\nThe method of gradient descent uses the opposite direction $-\\nabla f(a)$ to look for a minimum of $f$.\n\n## Explanations\n\n### L1 human\n\nIf you stand on a hillside and want to climb as fast as possible, walk in the direction of the gradient.\nIf you walk at right angles to the gradient, your height does not change at first: you follow a contour line.\n",
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