Calc Extrema

Front

What is the definition of a local extremum?

Back

A local extremum is a point in the domain of a function where the function takes on a maximum or minimum value relative to its immediate surroundings.

Front

What does it mean for a function to be increasing?

Back

A function f(x) is increasing on an interval if for any two points x1 and x2 in that interval, if x1 < x2, then f(x1) < f(x2).

Front

What is the relationship between f(x) and f'(x) regarding increasing and decreasing intervals?

Back

If f'(x) > 0, then f(x) is increasing. If f'(x) < 0, then f(x) is decreasing.

Front

How do you determine the intervals where a function is increasing or decreasing using a sign chart?

Back

A sign chart shows the sign of f'(x) over different intervals. Positive values indicate increasing intervals, while negative values indicate decreasing intervals.

Front

What is a critical point?

Back

A critical point occurs where f'(x) = 0 or where f'(x) is undefined. These points are potential locations for local extrema.

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 analyzing the sign of f'(x) before and after the critical point.

Front

What is a local maximum?

Back

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

Front

What is a local minimum?

Back

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

Front

How can you find the x-values where a function has a minimum using its derivative?

Back

Set f'(x) = 0 and solve for x to find critical points. Then use the first derivative test to determine if these points are minima.

Front

What does it mean for a function to have no extrema?

Back

A function has no extrema if it is either strictly increasing or strictly decreasing over its entire domain.