3-2 Critical Numbers and Extrema

Front

What is a critical number in calculus?

Back

A critical number, c, of a function f is a number in the domain of f such that f'(c) = 0 or f'(c) is undefined.

Front

How do you find critical numbers of a function?

Back

Critical numbers occur when f'(x) = 0 or where f'(x) is undefined.

Front

What does it mean if a function has no critical numbers?

Back

It means that the derivative f'(x) is never zero and is defined for all x in the domain.

Front

What is the first derivative test?

Back

The first derivative test is used to determine whether a critical point is a local maximum, local minimum, or neither by analyzing the sign of f'(x) around the critical point.

Front

What is the second derivative test?

Back

The second derivative test is used to determine the concavity of a function at a critical point to classify it as a local maximum or minimum.

Front

What is an increasing interval of a function?

Back

An increasing interval is a range of x-values where the function f(x) is rising, meaning f'(x) > 0.

Front

What is a decreasing interval of a function?

Back

A decreasing interval is a range of x-values where the function f(x) is falling, meaning f'(x) < 0.

Front

How do you determine the intervals of increase and decrease for a function?

Back

To determine intervals of increase and decrease, find the critical numbers, test intervals around those numbers, and analyze the sign of f'(x).

Front

What is a local maximum?

Back

A local maximum is a point where the function value is higher than all nearby points, typically found at critical points where f'(x) changes from positive to negative.

Front

What is a local minimum?

Back

A local minimum is a point where the function value is lower than all nearby points, typically found at critical points where f'(x) changes from negative to positive.