Concavity

Front

For a function g(x), if g''(3)=-8, what does this indicate about g(x) at x=3?

Back

g(x) is concave down at x=3.

Front

If (a,b) is a local maximum, what can be said about f''(a)?

Back

f''(a) is negative.

Front

What does concavity refer to in a function?

Back

Concavity refers to the direction of the curvature of the graph of a function.

Front

What describes the slope of a function?

Back

The slope of a function is described by its first derivative.

Front

What are the intervals on which the graph of f is concave down?

Back

The graph of f is concave down on the intervals -2 < x < -1 and 1 < x < 3.

Front

What does it mean if a function is concave up?

Back

A function is concave up if its second derivative is positive.

Front

What does it mean if a function is concave down?

Back

A function is concave down if its second derivative is negative.

Front

How can you determine the concavity of a function using its second derivative?

Back

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

Front

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

Back

The first derivative indicates the slope, while the second derivative indicates the concavity of the function.

Front

What is a point of inflection?

Back

A point of inflection is where the concavity of a function changes.