If a function $f$ is continuous on a closed interval $[a, b]$, then $f$ has both a maximum and a minimum value on that interval.

To guarantee the existence of absolute extrema for continuous functions on closed intervals, allowing us to find maximum and minimum values.

When the function is not continuous on the closed interval, or when the interval is not closed (open or half-open interval).

It assures us that a maximum and minimum exist, so we can use calculus techniques (finding critical points) to locate them.

It helps in finding the maximum profit, minimum cost, or optimal design within a given set of constraints.

To find the absolute maximum and minimum values of a continuous function on a closed interval.

The function must be continuous on a closed interval.

Find all critical points in the interval.

Evaluate the function at the critical points and endpoints.

The largest value is the absolute maximum, and the smallest value is the absolute minimum.

Critical points occur where $f'(x) = 0$ (x-intercepts) or where $f'(x)$ is undefined (vertical asymptotes, discontinuities).

$f(x)$ is increasing where $f'(x) > 0$ (above the x-axis) and decreasing where $f'(x) < 0$ (below the x-axis).

Local maxima occur at peaks, and local minima occur at valleys. Look for points where the graph changes direction.

Visually inspect the graph on the given interval. The highest point is the absolute maximum, and the lowest point is the absolute minimum.

Inflection points occur where $f''(x)$ changes sign (crosses the x-axis). These points indicate where the concavity of $f(x)$ changes.

Positive $f'(x)$ indicates $f(x)$ is increasing; negative $f'(x)$ indicates $f(x)$ is decreasing; $f'(x) = 0$ indicates a critical point.

Positive $f''(x)$ indicates $f(x)$ is concave up; negative $f''(x)$ indicates $f(x)$ is concave down.

It indicates that $f'(x) = 0$ at that point, suggesting a critical point (potential local max or min).

Concave up intervals look like a smile, while concave down intervals look like a frown. Note the points where the concavity changes (inflection points).

The absolute maximum is the highest point on the graph, and the absolute minimum is the lowest point on the graph.

First Derivative Test: Uses the sign change of $f'(x)$ to determine extrema. | Second Derivative Test: Uses the sign of $f''(x)$ at critical points to determine extrema.

Local Extrema: Maxima/minima within a specific interval. | Global Extrema: Absolute max/min over the entire domain.

Critical Points: Where $f'(x) = 0$ or is undefined (potential extrema). | Inflection Points: Where $f''(x)$ changes sign (change in concavity).

Open Interval: No guarantee of extrema; endpoints not included. | Closed Interval: Extreme Value Theorem guarantees extrema; endpoints must be checked.

Differentiability: Function has a derivative at a point (smooth curve). | Continuity: Function has no breaks, jumps, or holes.

The first derivative gives the rate of change of the function while the second derivative gives the rate of change of the first derivative.

Extrema are the maximum and minimum values of a function while critical points are the points where the derivative is zero or undefined.

A local extrema is the minimum or maximum in a specific interval while global extrema are the absolute minimum or maximum value of the function.

A maximum is the highest point in a given interval, while a minimum is the lowest point in a given interval.

A local maximum is the highest point in a given interval, while a local minimum is the lowest point in a given interval.