Analytical Applications of Differentiation

The function is increasing in that interval.

The Second Derivative Test cannot be applied.

The function is decreasing in that interval.

The function has a relative minimum in that interval.

The function has a local minimum at $x = c$.

The function has a local maximum at $x = c$.

The concavity of the function cannot be determined at $x = c$.

The function has an inflection point at $x = c$.

A point of discontinuity.

A relative minimum.

An inflection point.

A relative maximum.

Twice differentiable.

Concave upward.

Increasing.

Continuous.

The x-coordinate of inflection points.

The concavity and nature of the extrema.

The absolute extrema of a function.

The slope of the tangent line at critical points.

No conclusion can be drawn because more information is needed.

It is an inflection point where concavity changes.

It is a local minimum.

It is a local maximum.

There must be an error in calculation as this result isn't possible for extrema tests.

It confirms you have found either an extremum or an inflection point but does not indicate which one it could be.

The test is inconclusive; use another method to determine extrema or points of inflection.

You have found neither extremum nor inflection; move on without further analysis.

Find values where only second-derivative test fails; use first-derivative test instead.

Find values where $f'(x) = 0$ or where $f'(x)$ does not exist; then test with $f''(x)$.

Only find values where both $f'(x)$ and $f''(x)$ are equal to zero simultaneously.

Find values where only $f''(x) = 0$; ignore $f'(x)$.

Constant.

Increasing.

Concave upward.

Concave downward.

The test is inconclusive since only first derivative information can confirm extrema, not the second derivative.

There is a local maximum at $x = c$ because the second derivative being greater than zero indicates concavity downwards.

There is a local minimum at $x = c$ because the second derivative being greater than zero indicates concavity upwards.

There is no extremum at $x = c$ because the second derivative test cannot determine extrema conclusively.