Concavity

Front

What is concavity in the context of a function?

Back

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

Front

How is the concavity of a function determined?

Back

The concavity of a function is determined by its second derivative. If the second derivative is positive, the function is concave up; if negative, it is concave down.

Front

What does it mean if g''(x) > 0?

Back

If g''(x) > 0, the function g(x) is concave up at that point.

Front

What does it mean if g''(x) < 0?

Back

If g''(x) < 0, the function g(x) is concave down at that point.

Front

What is the relationship between the first and second derivatives in determining concavity?

Back

The first derivative indicates the slope of the function, while the second derivative indicates the rate of change of that slope, which determines concavity.

Front

If a function is concave up, what can be said about its first derivative?

Back

If a function is concave up, its first derivative is increasing.

Front

If a function is concave down, what can be said about its first derivative?

Back

If a function is concave down, its first derivative 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

How can you find inflection points using the second derivative?

Back

To find inflection points, set the second derivative equal to zero and solve for x. Check for sign changes in the second derivative around those points.

Front

What does it mean if g''(3) = -8?

Back

It means that the function g(x) is concave down at x = 3.