<aside> 📎 Open interval: $I_0 = (a,b)$

Closed interval: $I_0 = [a,b]$, contains endpoints

Neither: $I_0 = (a,b]$

</aside>

Absolute Extrema

• $f$ has an absolute maximum at the point $x = c$ if $f© ≥ f(x)$ for all $x \in dom f$
• $f$ has an absolute minimum at the point $x = c$ if $f(c) <= f(x)$ for all $x \in dom f$

General word for maxima or minima is extrema, absolute extrema are known as global extrema

Local Extrema

• $f$ has a local maximum at the point $x = c$ if there exists a small open interval $I_0$ around $c$ contained in dom $f$ such that $f(c ) ≥ f(x)$ for all $x \in I_0$
• $f$ has a local minimum at the point $x = c$ if there exists a small open interval $$I_0$ around $c$ contained in $dom f$ such that $f(c ) ≤ f(x)$ for all $x \in I_0$

Every global extrema is also a local extrema, but not the other way around

Global extrema may occur at the endpoints of a closed interval, but local extrema cannot

Critical Points

<aside> ❗ A function $f(x)$ has a critical point $x = c$, if there is a small open interval $I_0$ around $c$, where $I_0 \in dom f$, and $f’(c ) = 0$ or $f'(c )$ is not defined

</aside>

The First Derivative Test

• If $f’$ changes sign from positive when $x < c$ to negative when $x> c$, then $f(c )$ is a local maximum of $f$
• If $f’$ changes sign from negative when $x <c$ to positive when $x > c$, then $f(c )$ is a local minimum of $f$
• If $f’$ has the same sign for $x < c$ and $x > c$ then $f(c )$ is neither a local maximum nor a local minimum of $f$