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.

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.

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.

The absolute maximum and minimum values of a function over its entire domain.

Maximum or minimum values of a function within a specific region or interval of its domain.

A value $x = c$ in the domain of a function where $f'(c) = 0$ or $f'(c)$ is undefined.

The function has no breaks, jumps, or undefined points within the specified interval.

The set of all possible input values (x-values) for which the function is defined.

An interval that includes its endpoints, denoted as $[a, b]$.

A function is differentiable at a point if its derivative exists at that point.

The instantaneous rate of change of a function with respect to its variable.

The set of all possible output values (y-values) of the function.