Multivariable Calculus 18 | Local Extrema [dark version]

Local extrema in multivariable calculus are defined by comparing function values only inside a small neighborhood around a specific point, not across the whole domain. A function f has a local maximum at x0 if there exists an ε-neighborhood around x0 such that f(x0) is greater than or equal to f(x) for every x in the domain within that neighborhood. The “local” part means the comparison is restricted to points near x0, while the “maximum” part means x0 beats (or ties) all nearby values. To avoid flat cases like constant functions, the notion of an isolated local maximum strengthens the condition to a strict inequality: f(x0) is strictly greater than f(x) for all nearby points except x0 itself. Local minima mirror this definition with the inequality reversed, and a local extremum means either a local maximum or a local minimum.

Once local extrema are defined, the calculus criteria start to look like the one-variable story, but with gradients and Hessians. For continuously differentiable functions on R^n, any local extremum at x0 must be a critical point: the gradient ∇f(x0) must equal the zero vector. Intuitively, the gradient points in the direction of fastest increase, so at a local extremum there can’t be any nearby direction that increases the function.

To decide whether a critical point is actually a maximum or minimum, the discussion turns to Taylor’s theorem with a quadratic approximation using the Hessian matrix Hf, built from all second partial derivatives. Near a critical point x0, the function behaves like f(x0 + h) ≈ f(x0) + (quadratic form involving Hf) + a remainder term. The sign pattern of this quadratic form determines the type of extremum: - If Hf is positive definite, the quadratic form is strictly positive for every nonzero direction h, forcing f(x0 + h) to be larger than f(x0). That guarantees an isolated local minimum at x0. - If Hf is negative definite, the quadratic form is strictly negative for every nonzero h, forcing f(x0 + h) to be smaller than f(x0). That guarantees an isolated local maximum. - If Hf is indefinite—meaning there exist directions where the quadratic form is positive and other directions where it is negative—then nearby points can produce both larger and smaller values. This rules out any local extremum and produces a saddle point (the classic “looks like a max in one direction and a min in another” geometry).

The transcript also gives one-directional necessary conditions. If f has a local maximum at x0 (not necessarily isolated), then Hf must be negative semi-definite, allowing the quadratic form to be zero in some nonzero directions. Similarly, a local minimum forces Hf to be positive semi-definite. Because semi-definiteness can occur even when the extremum isn’t isolated, the strict “definite” conditions are sufficient for isolated extrema, while the semi-definite conditions are only necessary for extrema.

In short: local extrema are neighborhood-based comparisons; critical points require ∇f(x0)=0; and the Hessian’s definiteness—positive definite, negative definite, or indefinite—determines whether the point is an isolated minimum, isolated maximum, or a saddle point, with semi-definiteness providing the weaker necessary condition when isolation isn’t guaranteed.