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.

1. Find f'(x). 2. Determine critical points where f'(x) = 0 or is undefined. 3. Test intervals between critical points using f'(x) to determine if it's positive (increasing) or negative (decreasing).

1. Find critical points. 2. Test the sign of f'(x) to the left and right of each critical point. 3. If f'(x) changes sign, a local extremum exists.

1. Find f''(x). 2. Determine where f''(x) = 0 or is undefined. 3. Test intervals using f''(x) to determine if it's positive (concave up) or negative (concave down).

1. Find f''(x). 2. Solve for f''(x) = 0 or where it is undefined. 3. Verify that the concavity changes at those points.

1. Find critical points within the interval. 2. Evaluate f(x) at critical points and endpoints. 3. Compare values to find the absolute maximum and minimum.

1. Define the objective function. 2. Identify constraints. 3. Express the objective function in terms of one variable. 4. Find critical points. 5. Test for extrema.

1. Verify the function is continuous and differentiable on the given interval. 2. Calculate the average rate of change: (f(b)-f(a))/(b-a). 3. Find c such that f'(c) equals the average rate of change.

1. Find critical points. 2. Calculate the second derivative, f''(x). 3. Evaluate f''(x) at each critical point. 4. If f''(x) > 0, local minimum. If f''(x) < 0, local maximum.

Use the first derivative test (sign change of f'(x)) or the second derivative test (sign of f''(x)).

1. Identify critical points from f'(x). 2. Determine intervals of increasing/decreasing from f'(x). 3. Identify inflection points from f''(x). 4. Determine concavity from f''(x). 5. Sketch the graph using this information.