Analytical Applications of Differentiation

The function has a relative maxima at $x = 4$.

The function is incrementing on both sides.

The function's slope at $x = 4$ cannot be determined.

The function has a relative minima at $x = 4$.

No extremum

Overall minimum value of a function

Maximum value in a certain interval

Minimum value in a certain interval

Absolute extremes in isolated intervals where $f'(x) = 0$ but doesn't change sign.

Relative maximum points where $f'(x)$ changes from positive to negative.

Points of inflection where $f''(x) = 0$ or does not exist and $f'(x) = 0$.

Relative minimum point where $f'(x)$ changes from negative to positive.

The function is always increasing

The function is changing from decreasing to increasing

The function is changing from increasing to decreasing

The function is always decreasing

Point $b$ is a local minimum of $f(x)$.

Point $b$ is a local maximum of $f(x)$.

Point $b$ is an inflection point of $f(x)$.

Point $b$ has no significance to the graph of $f(x)$.

It implies an inflection point at $x = 3$.

It suggests a local minimum at $x = 3$.

It means there is no relative extrema at $x = 3$.

It indicates a local maximum at $x = 3$.

The original function has a local maximum at x0.

The original function is decreasing on this interval.

The original function is increasing on this interval.

The original function has an inflection point at x0.

There is likely a local minimum at $x = c$

There is likely a local maximum at $x = c$

There is neither a local maximum nor minimum at $x = c$

The behavior of function $g$ cannot be determined without further information.

It could be neither a maximum nor minimum since $f'(c)=0$ does not guarantee an extremum

It is likely a point of inflection since $f''(c) < 0$ signifies concavity change

It is likely a local maximum since the second derivative test indicates concave down

It is likely a local minimum because $f'(c)=0$ suggests so

To calculate the derivative of the function

To visually inspect the critical points

To find the relative extrema directly

To determine the intervals of increasing or decreasing