2nd Derivative Test

If f''(a) > 0, what can be concluded about the point (a, f(a))?

• It is a local minimum.

• It is a local maximum.

• It is an inflection point.

• It is a saddle point.

What is the significance of a local minimum in terms of the first and second derivatives?

• At a local minimum, f'(x) = 0 and f''(x) > 0.

• At a local minimum, f'(x) > 0 and f''(x) = 0.

• At a local minimum, f'(x) < 0 and f''(x) < 0.

• At a local minimum, f'(x) = 0 and f''(x) < 0.

What is the importance of the second derivative in optimization problems?

• It helps to identify the nature of critical points, which is essential for finding maximum and minimum values.

• It provides the slope of the function at a given point.

• It determines the concavity of the function only at endpoints.

• It is used to calculate the area under the curve.

What is the second derivative test for local extrema?

• If f''(c) > 0, then f has a local minimum at c; if f''(c) < 0, then f has a local maximum at c.

• If f''(c) = 0, then f has a local minimum at c; if f''(c) < 0, then f has a local maximum at c.

• If f''(c) > 0, then f has a local maximum at c; if f''(c) < 0, then f has a local minimum at c.

• If f''(c) > 0, then f is increasing at c; if f''(c) < 0, then f is decreasing at c.

If f''(a) < 0, what can be concluded about the point (a, f(a))?

• It is a local maximum.

• It is a local minimum.

• It is an inflection point.

• It is a point of discontinuity.

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

• The first derivative indicates the concavity of the function.

• The second derivative indicates the concavity of the function: f''(x) > 0 means concave up, f''(x) < 0 means concave down.

• The first derivative must be positive for the function to be concave up.

• The second derivative is irrelevant to concavity.

How can you determine if a function is concave up or concave down using the second derivative?

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

• If f''(x) = 0, the function is always concave up.

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

• If f''(x) < 0, the function is neither concave up nor down.

What does it mean if a function has no points of inflection?

• The function maintains the same concavity throughout its domain.

• The function has multiple local maxima and minima.

• The function is linear and does not change direction.

• The function oscillates between concave up and concave down.

How do you find the points of inflection for a function?

• Set the first derivative f'(x) equal to zero and solve for x.

• Set the second derivative f''(x) equal to zero and solve for x.

• Find the maximum and minimum points of the function.

• Evaluate the function at critical points.

What is the significance of a local maximum in terms of the first and second derivatives?

• At a local maximum, f'(x) = 0 and f''(x) < 0.

• At a local maximum, f'(x) > 0 and f''(x) > 0.

• At a local maximum, f'(x) = 0 and f''(x) = 0.

• At a local maximum, f'(x) < 0 and f''(x) > 0.