First and Second Derivative Tests Flashcard

Front

What does g''(x) < 0 indicate about the concavity of the function g(x)?

Back

The function g(x) is concave down.

Front

What is the relationship between the second derivative and the concavity of a function?

Back

The concavity of a function is described by its second derivative.

Front

If a function's first derivative is negative at a certain point, what does that imply about the function's behavior at that point?

Back

The function is decreasing at that point.

Front

What is the first derivative test used for?

Back

To determine local maxima and minima of a function.

Front

What is the second derivative test used for?

Back

To determine the concavity of a function and identify points of inflection.

Front

How do you find intervals of concavity for a function?

Back

By analyzing the sign of the second derivative over different intervals.

Front

What is the significance of a point where g''(x) = 0?

Back

It may indicate a point of inflection where the concavity changes.

Front

For the function y = x^3 + 6x^2, how do you determine where it is concave down?

Back

Find the second derivative and determine where it is negative.

Front

What does it mean if the first derivative of a function is zero at a point?

Back

It indicates a potential local maximum, minimum, or point of inflection.

Front

What is the average rate of change of a function over an interval?

Back

It is the change in the function's value divided by the change in the input value over that interval.