Second Derivative Test Practice

Front

What is the Second Derivative Test?

Back

The Second Derivative Test is a method used to determine the concavity of a function and to identify local maxima and minima by analyzing the sign of the second derivative, f''(x).

Front

How do you find critical numbers of a function?

Back

Critical numbers occur when f'(x) = 0 or where f'(x) is undefined.

Front

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

Back

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

Front

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

Back

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

Front

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

Back

The first derivative indicates where the function is increasing or decreasing, while the second derivative indicates the concavity, helping to confirm whether a critical point is a local maximum or minimum.

Front

What is a local maximum?

Back

A local maximum is a point where the function value is higher than all nearby points, typically found where f'(x) = 0 and f''(x) < 0.

Front

What is a local minimum?

Back

A local minimum is a point where the function value is lower than all nearby points, typically found where f'(x) = 0 and f''(x) > 0.

Front

How do you determine intervals of concavity?

Back

To determine intervals of concavity, find the second derivative f''(x) and analyze its sign (positive or negative) over different intervals.

Front

What does it mean if a function is concave up?

Back

A function is concave up if its graph opens upwards, indicating that the slope of the tangent line is increasing.

Front

What does it mean if a function is concave down?

Back

A function is concave down if its graph opens downwards, indicating that the slope of the tangent line is decreasing.