1. Find the derivative $f'(x)$. 2. Set $f'(x) = 0$ and solve for $x$. 3. Find where $f'(x)$ is undefined. 4. The solutions are critical points.

1. Find all critical points in the interval. 2. Evaluate the function at the critical points and endpoints. 3. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

1. Find the critical point $c$. 2. Check the sign of $f'(x)$ to the left and right of $c$. 3. If $f'(x)$ changes from positive to negative at $c$, then $c$ is a local maximum. 4. If $f'(x)$ changes from negative to positive at $c$, then $c$ is a local minimum.

1. Find the critical point $c$. 2. Find the second derivative $f''(x)$. 3. Evaluate $f''(c)$. 4. If $f''(c) > 0$, then $c$ is a local minimum. 5. If $f''(c) < 0$, then $c$ is a local maximum. 6. If $f''(c) = 0$, the test is inconclusive.

1. Verify that the function is continuous on the closed interval. 2. Find all critical points within the interval. 3. Evaluate the function at the critical points and the endpoints of the interval. 4. The largest and smallest values are the absolute maximum and minimum, respectively.

1. Identify points where $f'(x) = 0$ (x-intercepts). 2. Identify points where $f'(x)$ is undefined (e.g., vertical asymptotes, holes). 3. These x-values are the critical points of $f(x)$.

1. Visually inspect the graph. 2. Identify the highest point (absolute maximum) and the lowest point (absolute minimum) on the given interval. 3. Note the corresponding x and y values.

1. Find the second derivative $f''(x)$. 2. Determine where $f''(x) = 0$ or is undefined. 3. Check if $f''(x)$ changes sign at these points. If it does, it's an inflection point.

1. Check if the function is continuous. 2. Check if the interval is closed (i.e., includes its endpoints). If both conditions are met, the EVT applies.

1. Find the derivative and critical points. 2. Evaluate the function at critical points and endpoints. 3. Compare values to find absolute max/min.

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.

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.