Applying the second derivative test to a table of values

The video tutorial explores the concepts of local maxima and minima, focusing on the limitations of the first derivative test and the application of the second derivative test. It emphasizes the importance of critical points in determining concavity and highlights the need for caution when analyzing slopes far from the point of interest. The tutorial provides a detailed explanation of how to use the second derivative to assess concavity and identify critical points, offering a comprehensive understanding of these mathematical concepts.

What does the first derivative test indicate when the slope changes from positive to negative?

• No change in concavity

• A point of inflection

• A local maximum

• A local minimum

Why might the first derivative test not be reliable when using points far from the point of interest?

• The points might be too close

• The function might not be differentiable

• The slope might change unexpectedly

• The function might not be continuous

What does a positive second derivative at a point indicate about the graph?

• The graph has a point of inflection

• The graph has a local maximum

• The graph is concave down

• The graph is concave up

What must be true for the second derivative test to be applicable at a point?

• The first derivative must be zero

• The first derivative must be positive

• The function must be non-differentiable

• The second derivative must be zero

Why can't concavity be checked at non-critical points using the second derivative test?

• The second derivative is undefined

• The first derivative is not zero

• The function is not continuous

• The second derivative is zero