Extrema

Local Extrema

Definition: Local Minimum

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

We say that $f$ has a local minimum at $a\in \mathbb{D}$ if there is some $\epsilon >0$ such that

$$f(a)\le f(x)\forall x\in (a-\epsilon ;a+\epsilon )$$

This local minimum is $f(a)$.

Definition: Local Maximum

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

We say that $f$ has a local maximum at $a\in \mathbb{D}$ if there is some $\epsilon >0$ such that

$$f(a)\ge f(x)\forall x\in (a-\epsilon ;a+\epsilon )$$

This local maximum is $f(a)$.

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 $f:D\to \mathbb{R}$ be a real function.

We say that $f$ has a global minimum at $a\in D$ if

$$f(a)\le f(x)\forall x\in D$$

This global minimum is $f(a)$.

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 $f:D\to \mathbb{R}$ be a real function.

We say that $f$ has a global maximum at $a\in D$ if

$$f(a)\ge f(x)\forall x\in D$$

This global maximum is $f(a)$.

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.

Finding Extrema

Theorem: Critical Points and Local Extrema

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

All local extrema of $f$ occur at critical points of $f$.

PROOF

TODO

Theorem: Criteria of the First Derivative

If $f$ is continuous and differentiable at a critical point $x_0$, then:

• $f$ has a local minimum at $x_0$ if there is some $\epsilon >0$ such that

$$f^{\prime }(x)<0\forall x\in (x_0-\epsilon ;x_0)\text{ and }{f}^{\prime }(x)>0\forall x\in (x_0;x_0+\epsilon )$$

• $f$ has a local maximum at $x_0$ if there is some $\epsilon >0$ such that

$$f^{\prime }(x)>0\forall x\in (x_0-\epsilon ;x_0)\text{ and }{f}^{\prime }(x)<0\forall x\in (x_0;x_0+\epsilon )$$

PROOF

TODO

Theorem: Criteria of the Second Derivative

If $f$ is continuous and twice differentiable at a critical point $x_0$, then:

• $f$ has a local minimum at $x_0$ if its second derivative at $x_0$ is positive, i.e. $f^{\prime \prime }(x_0)>0$;

• $f$ has a local maximum at $x_0$ if its second derivative at $x_0$ is negative, i.e. $f^{\prime \prime }(x_0)<0$.

WARNING

If $f^{\prime \prime }(x_0)=0$ 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

Requirements: $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is continuous.

1. Determine the critical points $x_1,x_2,\cdots ,x_n\in D$ of $f$ by solving $f^{\prime }(x)=0$ and also seeing where $f$ is not differentiable.
2. Use the above criteria to check at which critical points $f$ has local extrema.
3. Evaluate $f$ at the places of its local extrema to obtain the values of the local minima and local maxima.
4. 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.

5. 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 $f$;