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.

Guarantees the existence of absolute max and min values for continuous functions on closed intervals, providing a basis for optimization problems.

Critical points are potential locations for local maxima and minima; they must be examined to determine if they are indeed extrema.

Discontinuities can lead to functions without a maximum or minimum value on a closed interval, violating the theorem's conditions.

Local extrema are maximum or minimum values within a specific interval, while global extrema are the absolute maximum and minimum values over the entire domain.

Critical points are the possible locations where a function can have a local maximum or minimum. They are found where the derivative is zero or undefined.

Yes, a critical point can be a point where the derivative is zero, but the function does not change direction (e.g., an inflection point with a horizontal tangent).

The global maximum or minimum can occur at an endpoint, even if the derivative is not zero or undefined there.

The first derivative test helps identify local extrema. If the derivative changes sign at a critical point, it indicates a local max or min.

The second derivative test can determine if a critical point is a local maximum or minimum. A positive second derivative indicates a local minimum, and a negative second derivative indicates a local maximum.

Local extrema are the minimum or maximum in a specific interval while absolute extrema are the absolute minimum or maximum value of the function.

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.