Extrema and Concavity

Front

What is a local minimum in calculus?

Back

A point (a,b) on the graph of a function f(x) where f(a) is less than f(x) for all x in some interval around a.

Front

What does f'(a) equal at a local minimum?

Back

f'(a) = 0.

Front

What does the sign chart for f'(x) indicate?

Back

It shows where the function is increasing or decreasing and helps identify local maxima and minima.

Front

What can be concluded if there are no extrema on a sign chart?

Back

The function does not change direction; it is either always increasing or always decreasing.

Front

What is concavity in relation to a function?

Back

Concavity describes the direction of the curvature of the graph of a function.

Front

What does it mean for a function to be concave up?

Back

The graph of the function opens upwards, and the second derivative f''(x) > 0.

Front

What does it mean for a function to be concave down?

Back

The graph of the function opens downwards, and the second derivative f''(x) < 0.

Front

What is an inflection point?

Back

A point on the graph where the concavity changes, typically where f''(x) = 0.

Front

How can you determine intervals of concavity from a derivative graph?

Back

By analyzing the sign of the second derivative f''(x) or the slope of f'(x).

Front

What is the relationship between f'(x) and local extrema?

Back

Local extrema occur where f'(x) = 0 or is undefined.