Extrema
Definition and necessary condition for local extrema.
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Definition 1. For \(D\subseteq\mathbb{R}\) and function \(f: D\to\mathbb{R}\): \(x_0\in D\) is called
local maximum if there exists a neighbourhood \(U\) around \(x_0\) such that \(f(x)\leq f(x_0)\) for \(x\in U\)
local minimum if there exists a neighbourhood \(U\) around \(x_0\) such that \(f(x)\geq f(x_0)\) for \(x\in U\)
local extremum if it is a local maximum or a local minimum.
global maximum if \(f(x)\leq f(x_0)\) for all \(x\in D\).
global minimum if \(f(x)\geq f(x_0)\) for all \(x\in D\).
Neighbourhood around \(U\) = (arbitrarily small) open subinterval of \(D\) that contains \(x_0\).
Theorem 2. For \(f: D\to \mathbb{R}\) differentiable, \(x_{0}\in D\) interior point:
If \(f\) has a local extremum at \(x_{0}\), then \(f'(x_{0})=0\).
For local extrema at boundary points \(x_0\) we can have \(f'(x_0)\neq 0\).
Definition 3. We call \(x_0\in D\) a stationary point if \(f'(x_0) = 0\).
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