2nd Derivative Test

It is a local minimum.

It is a local maximum.

It is an inflection point.

It is a saddle point.

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.

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.

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.

It is a local maximum.

It is a local minimum.

It is an inflection point.

It is a point of discontinuity.

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.

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.

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.

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.

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.