Local: Extrema within a specific interval. Global: Extrema over the entire domain.

First Derivative: Uses the sign of f'(x) to determine increasing/decreasing and local extrema. Second Derivative: Uses the sign of f''(x) to determine concavity and local extrema.

Concave Up: f''(x) > 0, curve opens upwards. Concave Down: f''(x) < 0, curve opens downwards.

Critical Points: f'(x) = 0 or undefined, potential local extrema. Inflection Points: f''(x) changes sign, change in concavity.

Minimization: Finding the minimum value of a function. Maximization: Finding the maximum value of a function.

Graphical: Sketching the graph to find extrema. Analytical: Using calculus (derivatives) to find extrema.

f'(x): Determines increasing/decreasing intervals and local extrema. f''(x): Determines concavity and inflection points.

MVT: Guarantees a point where the instantaneous rate of change equals the average rate of change. EVT: Guarantees the existence of a maximum and minimum value on a closed interval.

Relative: Local maximum or minimum within a specific interval. Absolute: Global maximum or minimum over the entire domain.

Function: Represents the original relationship between x and y. Derivative: Represents the rate of change of the function.

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

$f'(c) = \frac{f(b) - f(a)}{b - a}$ 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.

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.