Analytical Applications of Differentiation

There are positive values for $f''(x)$.

$f''(x)$ must equal zero.

There are negative values for $f''(x)$.

The sign of $f''(x)$ cannot be determined by observing.

An inflection point is always a maximum or minimum point

Inflection points occur where the second derivative is zero

A function can have multiple inflection points

Inflection points occur where the first derivative is zero

C is neither an extremum nor an inflection point since both derivatives are zero simultaneously.

C must be an absolute minimum since both derivatives are non-negative.

C must be a relative maximum because both first and second derivatives equal zero.

It cannot be determined whether c is an extremum or inflection point without further analysis.

The function is concave up

The function has a maximum point

The function is concave down

The function has an inflection point

The function has reached its maximum value.

The first derivative at this point must also be zero.

It indicates that the function is neither increasing nor decreasing.

It could be an inflection point where concavity changes.

The concavity cannot be determined with this information

The function is concave up

The function has an inflection point

The function is concave down

The function is concave up everywhere except possibly at $x=c$.

The function is increasing for all real numbers.

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

The function has no points of inflection.

The curve will have a downward opening curvature near x=c.

The curve will exhibit linear behavior near x=c.

The curve will have neither upward nor downward opening curvature near x=c.

The curve will cross its tangent line at x=c.

Both $h'$ and $h''$ are non-zero within region R.

There’s exactly one point of inflection in region R while $h'$ can be zero there momentarily.

Region R contains neither relative extrema nor points of inflection necessarily.

A point where the concavity of a function changes

A point where the first derivative is zero

A point where the function has a maximum or minimum value

A point where the second derivative is zero