If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

If f(x) is continuous on [a, b], then f(x) attains both a maximum and a minimum value on that interval.

It is used to relate the average rate of change of a function over an interval to its instantaneous rate of change at some point within the interval.

It guarantees the existence of absolute maximum and minimum values for continuous functions on closed intervals, which is crucial for optimization problems.

Function must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b).

Function must be continuous on the closed interval [a, b].

Rolle's Theorem is a special case of the Mean Value Theorem where f(a) = f(b).

It ensures that a continuous function on a closed interval has a maximum and minimum value, allowing us to find the optimal solution.

It states that at some point in an interval, the derivative of the function is equal to the average rate of change over that interval.

It guarantees that a continuous function on a closed interval will have both an absolute maximum and an absolute minimum within that interval.

If a function is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

If a function f(x) is continuous on a closed interval [a, b], then f(x) must attain both a maximum and a minimum value on that interval.

A value c in the domain of f(x) such that either f'(c) = 0 or f'(c) does not exist.

The highest and lowest points of a function over its entire domain.

The highest and lowest points of a function over a specific subinterval of its domain.

The curvature of a function at a given point; indicates whether the function is 'bending up' or 'bending down'.

A point on a curve where the concavity changes.

Mathematical problems that involve finding the best solution (minimum or maximum) among a set of possible solutions.

A method used to determine whether a function is increasing or decreasing on a specific interval by analyzing the sign of its first derivative.

A method used to determine the absolute extrema of a continuous function on a closed interval by evaluating the function at critical points and endpoints.

$\frac{f(b) - f(a)}{b - a}$

$f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $c \in (a, b)$

Solve $f'(x) = 0$ or find where $f'(x)$ is undefined.

$f''(x) > 0$ implies concave up; $f''(x) < 0$ implies concave down.

Solve $f''(x) = 0$ or find where $f''(x)$ is undefined, and verify concavity changes.

f(x) is increasing.

f(x) is decreasing.

Possible inflection point.

If $f'(c) = 0$ and $f''(c) > 0$, then $f(c)$ is a local minimum. If $f'(c) = 0$ and $f''(c) < 0$, then $f(c)$ is a local maximum.

1. Define the objective function. 2. Identify constraints. 3. Find critical points. 4. Test for extrema.