Extrema and Increasing Decreasing Intervals

Front

What is an extremum in a function?

Back

An extremum is a point in the domain of a function where the function takes on a maximum or minimum value. It can be classified as a local (or relative) extremum or a global (or absolute) extremum.

Front

How do you determine the increasing intervals of a function?

Back

A function is increasing on an interval if, for any two points in that interval, the function value at the second point is greater than the function value at the first point. This can be determined by analyzing the first derivative of the function.

Front

What is a decreasing interval in a function?

Back

A function is decreasing on an interval if, for any two points in that interval, the function value at the second point is less than the function value at the first point. This can be determined by analyzing the first derivative of the function.

Front

How can you find the extrema of a function?

Back

To find the extrema of a function, you can use the first derivative test. Set the first derivative equal to zero to find critical points, then use the second derivative test or analyze the sign of the first derivative around those points.

Front

What is the first derivative test?

Back

The first derivative test is a method used to determine whether a critical point is a local maximum, local minimum, or neither by examining the sign of the first derivative before and after the critical point.

Front

What is the second derivative test?

Back

The second derivative test is a method used to determine the concavity of a function at a critical point. If the second derivative is positive at a critical point, it indicates a local minimum; if negative, a local maximum.

Front

Define local maximum and local minimum.

Back

A local maximum is a point where the function value is greater than the values of the function at nearby points. A local minimum is a point where the function value is less than the values of the function at nearby points.

Front

What is a global maximum and global minimum?

Back

A global maximum is the highest point over the entire domain of the function, while a global minimum is the lowest point over the entire domain.

Front

How do you identify the number of extrema in a graph?

Back

The number of extrema in a graph can be identified by counting the peaks (maxima) and valleys (minima) in the graph.

Front

What is the significance of critical points in a function?

Back

Critical points are significant because they are potential locations for local maxima and minima, and they occur where the first derivative is zero or undefined.