Concavity and Extrema

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 (like a frown).

Front

What does f''(x) > 0 indicate about the function f(x)?

Back

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

Tags

CCSS.HSF.IF.B.4

Front

What does f''(x) < 0 indicate about the function f(x)?

Back

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

Tags

CCSS.HSF.IF.B.4

Front

What is a local maximum?

Back

A local maximum is a point on the graph of a function where the function value is higher than the values of the function at nearby points.

Front

What is a local minimum?

Back

A local 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

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

Back

When f'(x) changes from positive to negative, it indicates that there is a local maximum at that point.

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 there is a local minimum at that point.

Front

What is an inflection point?

Back

An inflection point is a point on the graph of a function where the concavity changes, which occurs when f''(x) = 0 or is undefined.

Tags

CCSS.HSF.IF.A.2

Front

What is the significance of critical points in a function?

Back

Critical points are values of x where f'(x) = 0 or f'(x) is undefined, and they are potential locations for local maxima and minima.

Front

How do you determine if a critical point is a maximum or minimum?

Back

To determine if a critical point is a maximum or minimum, you can use the first derivative test or the second derivative test.