Concavity Flashcard

Front

What is an inflection point?

Back

An inflection point is a point on the graph of a function where the concavity changes, which typically occurs when the second derivative, f''(x), is equal to zero.

Front

What does it mean when f'(x) changes from negative to positive?

Back

When f'(x) changes from negative to positive, it indicates that the function f(x) has a relative minimum at that point.

Front

How do you determine the intervals where a function is decreasing?

Back

A function is decreasing on intervals where its first derivative, f'(x), is less than zero.

Front

What is the relationship between the second derivative and concavity?

Back

If f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

Front

What is the significance of f''(x) = 0?

Back

When f''(x) = 0, it may indicate a potential inflection point, but further analysis is needed to confirm a change in concavity.

Front

Define concave up and concave down in terms of a function's graph.

Back

A function is concave up if its graph opens upwards (like a cup) and concave down if it opens downwards (like a frown).

Front

What is a relative minimum?

Back

A relative minimum is a point on the graph of a function where the function value is lower than the values of the function at nearby points.

Front

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

Back

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

Front

What does it mean for a function to be undefined at a point?

Back

A function is undefined at a point if there is no value of the function at that point, often due to division by zero or a discontinuity.

Front

What is the first derivative test?

Back

The first derivative test is a method used to determine whether a critical point is a relative maximum or minimum by analyzing the sign of f'(x) around that point.