Absolute Maximum and Minimum

Front

What is the definition of an absolute maximum of a function on an interval?

Back

An absolute maximum of a function on an interval is the highest value that the function attains within that interval.

Front

What is the definition of an absolute minimum of a function on an interval?

Back

An absolute minimum of a function on an interval is the lowest value that the function attains within that interval.

Front

How do you find the absolute extrema of a function on a closed interval?

Back

To find the absolute extrema, evaluate the function at critical points and endpoints of the interval, then compare the values.

Front

What is a critical point of a function?

Back

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

Front

What is the first derivative test for local extrema?

Back

The first derivative test states that if the derivative changes from positive to negative at a critical point, there is a local maximum; if it changes from negative to positive, there is a local minimum.

Front

What is the second derivative test for local extrema?

Back

The second derivative test states that if the second derivative at a critical point is positive, the function has a local minimum; if negative, it has a local maximum.

Front

What is the difference between local and absolute extrema?

Back

Local extrema are the highest or lowest points in a small neighborhood, while absolute extrema are the highest or lowest points over the entire interval.

Front

What is a sign chart and how is it used in finding extrema?

Back

A sign chart is a graphical representation that shows where a function's derivative is positive, negative, or zero, helping to identify intervals of increase and decrease.

Front

What is the role of endpoints in finding absolute extrema?

Back

Endpoints must be evaluated along with critical points because absolute extrema can occur at the endpoints of the interval.

Front

How do you determine if a function has an absolute maximum or minimum?

Back

By comparing the values of the function at critical points and endpoints, the highest value is the absolute maximum and the lowest is the absolute minimum.