Maxima and minima are points on a graph where a function reaches its highest (maximum) or lowest (minimum) value. These points are also called extrema.

Local Maximum/Minimum: The highest/lowest point in a small interval around a point. Absolute Maximum/Minimum: The highest/lowest point over the entire domain of the function.

To find local maxima and minima: 1. Find the first derivative of the function f'(x). 2. Set the first derivative to zero and solve for x (find critical points). 3. Use the second derivative f''(x) to determine if the critical points are maxima or minima: If f''(x) > 0, it's a local minimum. If f''(x) < 0, it's a local maximum.

The first derivative test involves: 1. Finding critical points by setting f'(x) = 0. 2. Checking the sign of f'(x) on either side of the critical points: If f'(x) changes from positive to negative, the point is a local maximum. If f'(x) changes from negative to positive, the point is a local minimum.

The second derivative test involves: 1. Finding the first derivative and setting it to zero to find critical points. 2. Using the second derivative to classify these points: If f''(x) > 0 at a critical point, it's a local minimum. If f''(x) < 0 at a critical point, it's a local maximum. If f''(x) = 0, the test is inconclusive.(Go back to first derivative for maxima or minima or higher order derivative test)

If the second derivative test fails (f''(c) = 0), use the first derivative test or higher-order derivatives to determine the nature of the critical point.

A point of inflection is where the function changes concavity, which means f''(x) changes sign or doesn't exist. It's not a maximum or minimum, but rather a point where the graph shifts from being concave up to concave down, or vice versa.