Extrema and Increasing Decreasing Intervals

An increasing interval is a range of x-values where the function's output (y-values) is rising as x increases.

A decreasing interval is a range of x-values where the function's output (y-values) is falling as x increases.

By analyzing the slope of the tangent line: if the slope is positive, the function is increasing; if negative, it is decreasing.

An extremum is a point in a function where it reaches a local maximum or minimum value.

A local maximum is the highest point in a neighborhood, while a local minimum is the lowest point in a neighborhood.

By finding critical points where the first derivative is zero or undefined, and then testing intervals around these points.

It means the function does not have any local maximum or minimum points within the given interval.

The second derivative indicates concavity: if positive, the function is concave up; if negative, concave down.

By analyzing the sign of the first derivative: positive indicates increasing, negative indicates decreasing.

Critical points are potential locations for extrema, where the first derivative is zero or undefined.