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.

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 concavity cannot be determined with this information

The function is concave up

The function has an inflection point

The function is concave down

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

The rate of change can only be determined by the first derivative

The concavity has no relationship with the rate of change

The concavity determines the speed at which the function is increasing or decreasing

The concavity determines whether the function is increasing or decreasing

The function is decreasing and concave up

The function is increasing and concave down

The function is decreasing and concave down

The function is increasing and concave up

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.

The interval $(-\infty,-2)$

The interval $(1,\infty)$

The interval $(-\infty,-\frac{8}{9})$

The interval $(-\infty,0)$

The second derivative of $g$, $g''(x)$, is less than zero on $(a,b)$.

Tangent lines become vertical indicating points of inflection at each end of interval $(a,b)$.

Function $g$ must have a local minimum somewhere within interval $(a,b)$.

The first derivative $g'(x)$ must be positive for all $x$ in $(a, b)$.

The graph must show a local extremum at every interval end.

The graph always shows constant slope at those intervals.

The graph may be transitioning between different types of concavity.

The graph necessarily displays a point of inflection at each interval.