Glossary

Critical Points

Criticality: 3

A point $x=c$ in the domain of a function where the derivative $f'(c)$ is either equal to zero or is undefined.

Example:

For $f(x) = x^3 - 3x$ , we find $f'(x) = 3x^2 - 3$ . Setting $f'(x)=0$ gives $x=\pm 1$ , so $x=1$ and $x=-1$ are critical points.

Extreme Value Theorem

Criticality: 3

A theorem stating that a continuous function on a closed interval [a,b] must attain both an absolute maximum and an absolute minimum value within that interval.

Example:

If a function $f(x)$ is continuous on the interval $[-2, 5]$ , the Extreme Value Theorem guarantees that it will have a highest and lowest point somewhere between $x=-2$ and $x=5$ .

Global Extrema

Criticality: 3

The absolute highest (global maximum) or lowest (global minimum) value a function attains over its entire domain or a specified interval.

Example:

For the function $f(x) = x^2$ on the interval $[-3, 2]$ , the global maximum is 9 (at $x=-3$ ) and the global minimum is 0 (at $x=0$ ).

Local Extrema

Criticality: 3

Points where a function reaches a maximum (local maximum) or minimum (local minimum) value within a specific, localized region or open interval around that point.

Example:

In a roller coaster's path, a small dip before a big climb could be a local minimum, even if it's not the lowest point on the entire ride.

Glossary

Critical Points

Criticality: 3

A point $x=c$ in the domain of a function where the derivative $f'(c)$ is either equal to zero or is undefined.

Example:

For $f(x) = x^3 - 3x$ , we find $f'(x) = 3x^2 - 3$ . Setting $f'(x)=0$ gives $x=\pm 1$ , so $x=1$ and $x=-1$ are critical points.

Extreme Value Theorem

Criticality: 3

A theorem stating that a continuous function on a closed interval [a,b] must attain both an absolute maximum and an absolute minimum value within that interval.

Example:

If a function $f(x)$ is continuous on the interval $[-2, 5]$ , the Extreme Value Theorem guarantees that it will have a highest and lowest point somewhere between $x=-2$ and $x=5$ .

Global Extrema

Criticality: 3

The absolute highest (global maximum) or lowest (global minimum) value a function attains over its entire domain or a specified interval.

Example:

For the function $f(x) = x^2$ on the interval $[-3, 2]$ , the global maximum is 9 (at $x=-3$ ) and the global minimum is 0 (at $x=0$ ).

Local Extrema

Criticality: 3

Points where a function reaches a maximum (local maximum) or minimum (local minimum) value within a specific, localized region or open interval around that point.

Example:

In a roller coaster's path, a small dip before a big climb could be a local minimum, even if it's not the lowest point on the entire ride.