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 $x = 4$.

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

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

p > / <Zero, q=r

p ≠ q ≠ r

p ≈ Zero irReleVant oF q,r

p > / <Zero, q ≠ r

If $a < 0$ and $f'(c) < 0$, then $g'(c) > 0$ indicating no extremum at $c$.

For all positive values of $a$, $g$ will have an extremum whenever $f$ has one.

If $a > 0$ and $f'(c) > 0$, then $g'(c) > 0$ indicating no extremum at $c$.

If $a > 0$ and $f'(c) \neq 0$, then $g$ has an extremum at $c$.

Both tests are equally useful, as they provide the same information about extrema at point $c$.

The Second Derivative Test conclusively determines that $f(c)$ is a point of inflection, hence its preference.

The First Derivative Test directly demonstrates that $f(c)$ is a relative maximum by showing an increase followed by a decrease in function values.

The Second Derivative Test requires less computation since it only involves evaluating the concavity at point $c$.

The function is changing from increasing to decreasing

The function is always decreasing

The function is changing from decreasing to increasing

The function is always increasing

Mean Value Theorem

Extreme Value Theorem

Second Derivative Test

Candidates Test

The graph continues to increase or decreases smoothly with no critical points or maxima occurring on the interval $(a,c)$.

The graph has a relative minimum at $b$.

There is a continuous increase to a maximum followed by a decrease from maximum to minimum at $b$.

The graph has a relative maximum at $b$.