At some point between two points on a curve, the instantaneous rate of change (derivative) equals the average rate of change (slope of the secant line).

A continuous function on a closed interval is guaranteed to have a maximum and minimum value within that interval.

By analyzing the sign change of f'(x) around a critical point, we can determine if the point is a local max (f' changes from + to -) or local min (f' changes from - to +).

The sign of the second derivative indicates the concavity of the function. Positive means concave up, negative means concave down.

Local extrema can only occur at critical points (where f'(x) = 0 or is undefined).

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

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

Evaluate the function at all critical points and endpoints within the interval. The largest value is the absolute maximum, and the smallest is the absolute minimum.

f'(x) indicates where f(x) is increasing/decreasing. f''(x) indicates the concavity of f(x).

To find the best possible solution (maximum or minimum) to a problem under given constraints.

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.