Concavity

Front

What is concavity in a function?

Back

Concavity refers to the direction in which a function curves. A function is concave up if its graph opens upwards (like a cup), and concave down if it opens downwards.

Front

What does it mean when f''(x) > 0?

Back

If f''(x) > 0, the function is concave up at that point, indicating that the slope of the tangent line is increasing.

Front

What does it mean when f''(x) < 0?

Back

If f''(x) < 0, the function is concave down at that point, indicating that the slope of the tangent line is decreasing.

Front

What is an inflection point?

Back

An inflection point is a point on the graph of a function where the concavity changes from concave up to concave down or vice versa.

Front

When does a function have a local maximum?

Back

A function has a local maximum at a point where f'(x) changes from positive to negative.

Front

When does a function have a local minimum?

Back

A function has a local minimum at a point where f'(x) changes from negative to positive.

Front

What does f'(x) = 0 indicate?

Back

f'(x) = 0 indicates a critical point, which could be a local maximum, local minimum, or an inflection point.

Front

How can you determine the number of extrema in a function?

Back

The number of extrema can be determined by analyzing the critical points and the behavior of the first derivative.

Front

What is the relationship between the first and second derivatives?

Back

The first derivative f'(x) indicates the slope of the function, while the second derivative f''(x) indicates the concavity of the function.

Front

What does it mean if f'(x) is negative?

Back

If f'(x) is negative, the function is decreasing at that point.

Tags

CCSS.HSF.IF.B.4