Analytical Applications of Differentiation

There are no critical points for $h(x)$ within $[a,b]$.

The second derivative test fails to determine concavity over $[a,b]$.

The function $h(x)$ has neither strictly increasing nor strictly decreasing intervals on $[a,b]$.

The derivative $h'(x)$ does not exist over $[a,b]$.

The function is increasing

The function is decreasing

The function has a local minimum

The function has a local maximum

By analyzing the sign changes of the derivative

By evaluating the function at critical points

By finding inflection points

By calculating the average rate of change

It creates a maximum at $x = c$, so $f(x)$ is not increasing over the entire interval.

It does not affect; $f(x)$ is still increasing on $(a, b)$.

It indicates that there is a minimum at $x = c$, creating a section where the function decreases.

The function has an inflection point at $x = c$, changing concavity but not monotonicity.

Between the critical points where $g'(x) = 0$ and changes from negative to positive

Between the critical points where $g'(x) = 0$ and changes from positive to negative

Wherever its second derivative $g''(x) < 0$

When its first derivative satisfies $g'(x) > -1$

It has a local maximum at point $a$.

It has a local minimum at point $a$.

The function is constant near point $a$.

It has an inflection point at point $a$.

The function likely maintains its direction of monotonicity through this critical point.

This would represent an undetected extremum due to computational error.

The critical point separates regions where $p(x)$'s derivative changes sign.

There must be an inflection point near this critical point shifting concavity.

The function is always increasing because the negative linear term only appears once so its effect is minimized

The function is always decreasing because the negative linear term appears twice so its effect is doubled

The function will decrease before the turning point and increase after the turning point since the exponential term is always negative

The function will increase before the turning point and decrease after the turning point since exponential growth dominates the negative effect of the linear term

It must be constantly equal to one for all values near $c$.

It must be negative and decreasing rapidly.

It must be positive and increasing rapidly.

It could be zero or undefined.

The derivative will be a large positive number.

The derivative will be a large negative number.

The derivative must equal pi.

The derivative will be zero or not exist.