Extrema

Front

What is an extremum?

Back

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

Front

What is the difference between relative and absolute extrema?

Back

Relative extrema are the highest or lowest points in a specific interval, while absolute extrema are the highest or lowest points over the entire domain of the function.

Front

How do you find relative maxima and minima?

Back

To find relative maxima and minima, take the derivative of the function, set it to zero, and solve for critical points. Then, use the second derivative test or the first derivative test to classify these points.

Front

What is the first derivative test?

Back

The first derivative test involves analyzing the sign of the derivative before and after a critical point to determine if it is a maximum or minimum.

Front

What is the second derivative test?

Back

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

Front

What is a global (absolute) maximum?

Back

A global maximum is the highest point over the entire domain of a function.

Front

What is a global (absolute) minimum?

Back

A global minimum is the lowest point over the entire domain of a function.

Front

What does it mean for a function to be increasing or decreasing?

Back

A function is increasing on an interval if its derivative is positive, and decreasing if its derivative is negative.

Front

What is a critical point?

Back

A critical point is a point in the domain of a function where the derivative is zero or undefined.

Front

How can you determine the number of extrema in a function?

Back

The number of extrema can be determined by finding the critical points and evaluating the function at these points and the endpoints of the interval.