Derivation — The Gradient as the Direction of Steepest Ascent

The claim that "the gradient points in the direction of steepest ascent" is not just a slogan — it has a clean proof using the Cauchy-Schwarz inequality. Similarly, the fact that "the gradient is perpendicular to level curves" follows from a one-line chain-rule argument. Both proofs are short but non-obvious the first time; understanding them makes the gradient intuition durable rather than memorised.

• prove that $D_\mathbf{u} f$ is maximised when $\mathbf{u} = \nabla f/\|\nabla f\|$ using Cauchy-Schwarz;
• prove that $\nabla f$ is perpendicular to the level curve $\{f = c\}$ at any point via the chain rule;
• interpret the shadow price $\lambda$ in Lagrange multipliers as the rate of change of the optimum with respect to the constraint — the financial interpretation of the gradient perpendicularity condition.

1. 1. Proof: The Gradient Is the Steepest Ascent Direction
2. 1. Proof: The Gradient Is Perpendicular to Level Curves
3. 1. The Lagrange Condition from Gradient Perpendicularity
4. 1. Directional Derivative at a Critical Point