Analytical Applications of Differentiation

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.

Knowing only the second derivative suffices for applying the second derivative test.

The sign of the first derivative shows a function is increasing or decreasing near a critical point, while the second sign indicates concavity or convexity at that point.

The first derivative sign shows the presence of an inflection point; the second sign determines if the inflection is a maximum or minimum.

The sign of the first derivative is the main indicator for potential extremal points regardless of the second.

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.

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.

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.

Twice differentiable.

Concave upward.

Increasing.

Continuous.

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.