First and Second Derivative Tests Flashcard

Front

What is the first derivative test used for?

Back

The first derivative test is used to determine the local maxima and minima of a function by analyzing the sign of the first derivative.

Front

What does it mean if f'(x) > 0 on an interval?

Back

If f'(x) > 0 on an interval, it means that the function f(x) is increasing on that interval.

Front

What does it mean if f'(x) < 0 on an interval?

Back

If f'(x) < 0 on an interval, it means that the function f(x) is decreasing on that interval.

Front

What is the second derivative test used for?

Back

The second derivative test is used to determine the concavity of a function and to identify local extrema.

Front

What does it indicate if f''(x) > 0 at a point?

Back

If f''(x) > 0 at a point, the function is concave up at that point, indicating a local minimum.

Front

What does it indicate if f''(x) < 0 at a point?

Back

If f''(x) < 0 at a point, the function is concave down at that point, indicating a local maximum.

Front

What does it mean if f''(x) = 0 at a point?

Back

If f''(x) = 0 at a point, the test is inconclusive; further analysis is needed to determine concavity.

Front

How do you find the intervals of concavity for a function?

Back

To find intervals of concavity, compute the second derivative and determine where it is positive (concave up) or negative (concave down).

Front

What is the relationship between the first and second derivatives in determining local extrema?

Back

The first derivative indicates increasing or decreasing behavior, while the second derivative indicates concavity, helping to confirm local extrema.

Front

What is a local maximum?

Back

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