Extrema of Real Functions#

Local Extrema#

Definition: Local Minimum

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function.

We say that \(f\) has a local minimum at \(p \in \mathcal{D}\) if there exists some open neighborhood \(N(p)\) such that

\[ f(p) \le f(x) \qquad \forall x \in N(p) \]

This local minimum is \(f(p)\).

Definition: Local Maximum

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function.

We say that \(f\) has a local maximum at \(p \in \mathcal{D}\) if there exists some open neighborhood \(N(p)\) such that

\[ f(p) \ge f(x) \qquad \forall x \in N(p) \]

This local maximum is \(f(p)\).

Definition: Local Extremum

The local minima and maxima of a function are known as its local extrema.

TODO: Add Diagram

Global Extrema#

Definition: Global Minimum

Let \(f: \mathcal{D} \to \mathbb{R}\) be a real function.

We say that \(f\) has a global minimum at \(p \in \mathcal{D}\) if

\[ f(p) \le f(x) \qquad \forall x \in \mathcal{D} \]

This global minimum is \(f(p)\).

Note

There cannot be more than one value for the global minimum, but there may be multiple places in the domain of the function where said minimum occurs.

Definition: Global Maximum

Let \(f: \mathcal{D} \to \mathbb{R}\) be a real function.

We say that \(f\) has a global maximum at \(p \in \mathcal{D}\) if

\[ f(p) \ge f(x) \qquad \forall x \in \mathcal{D} \]

This global maximum is \(f(p)\).

Note

There cannot be more than one value for the global maximum, but there may be multiple places in the domain of the function where said maximum occurs.

Definition: Global Extremum

The global minimum and maximum of a function are known as its global extrema.

Theorem: Global Extremum \(\implies\) Local Extremum

Every global minimum is a local minimum and every global maximum is a local maximum.

Proof

TODO

Finding Extrema#

Theorem: Critical Points and Local Extrema

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function.

If \(f\) has a local extremum at \(p \in \mathcal{D}\), then \(f\) has a critical point at \(p\).

Proof

If \(f\) is not differentiable at \(p\), then it trivially has a critical point at \(p\).

We need to prove two additional things:

(I) If \(f\) is differentiable at \(p\) and has a local minimum there, then \(f'(p) = 0\).
(II) If \(f\) is differentiable at \(p\) and has a local maximum there, then \(f'(p) = 0\).

Proof of (I):

Since \(f\) is differentiable at \(p\), for each \(\varepsilon \gt 0\), there exists some \(\delta \gt 0\) such that

\[ 0 \lt |h - p| \lt \delta \implies \left\vert\frac{f(p + h) - f(p)}{h} - f'(p)\right\vert \lt \varepsilon. \]

Since \(f\) has a local minimum at \(p\), there exists some \(\delta' \le \delta\) such that

\[ f(p) \le f(x) \]

for all \(x\) with \(0 \lt |x - p| \lt \delta'\). TODO

Theorem: Criteria of the First Derivative

If \(f\) is differentiable at a critical point \(x_0\), then:

\(f\) has a local minimum at \(x_0\) if there is some \(\varepsilon \gt 0\) such that

\[ f'(x) \lt 0 \qquad \forall x \in (x_0 - \varepsilon; x_0) \qquad \text{ and } \qquad f'(x) \gt 0 \qquad \forall x \in (x_0; x_0 + \varepsilon) \]

\(f\) has a local maximum at \(x_0\) if there is some \(\varepsilon \gt 0\) such that

\[ f'(x) \gt 0 \quad \forall x \in (x_0-\varepsilon; x_0)\qquad \text{ and } \qquad f'(x) \lt 0 \quad \forall x\in (x_0; x_0+\varepsilon) \]

Proof

TODO

Theorem: Criteria of the Second Derivative

If \(f\) is differentiable at a critical point \(x_0\), then:

\(f\) has a local minimum at \(x_0\) if \(f''(x_0) \gt 0\);
\(f\) has a local maximum at \(x_0\) if \(f''(x_0) \lt 0\).

Warning

If \(f''(x_0) = 0\) or \(f\) is not twice differentiable at \(x_0\), then \(f\) may or may not have a local extremum at \(x_0\), but we cannot use the second derivative to verify this.

Proof

TODO

Algorithm: Finding the Extrema of a Function

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function.

Determine the critical points \(x_1, x_2, \cdots, x_n \in \mathcal{D}\) of \(f\) by solving \(f'(x) = 0\) and also seeing where \(f\) is not differentiable.
Use the above criteria to check at which critical point \(f\) has local extrema.
Evaluate \(f\) at the places of its local extrema to obtain the values of the local minima and local maxima.
Evaluate \(f\) at the following locations:
- If \(D = [a;b]\) where \(a,b \in \mathbb{R}\), evaluate \(f(a)\) and \(f(b)\);
- If \(D = [a;b)\) where \(a \in \mathbb{R}\) and \(b \in \mathbb{R} \cup \{\infty\}\), evaluate \(f(a)\) and \(\lim_{x\to b} f(x)\);
- If \(D = (a;b]\) where \(a \in \mathbb{R} \cup \{-\infty \}\) and \(b \in \mathbb{R}\), evaluate \(\lim_{x\to a} f(x)\) and \(f(b)\);
- If \(D = (a;b)\) where \(a \in \mathbb{R} \cup \{-\infty \}\) and \(b \in \mathbb{R} \cup \{\infty\}\), evaluate \(\lim_{x\to a} f(x)\) and \(\lim_{x\to b} f(x)\);
- If \(D = D_1 \cup \cdots \cup D_n\) is a union of disjoint intervals \(D_1, \cdots, D_n\), then perform Step 4 separately for each interval.
Compare the local extrema of \(f\) with the values from Step 4:
- If there is a greatest value, then it is the global maximum of \(f\);
- If there is a smallest value, then it is the global minimum of \(f\);