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 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

The sign of ${g}'(x)$ does not change as $x$ passes through $c$.

${g}''(c) = 0$

${g}(c+1) > {g}(c-1)$

${g}'(c) > 0$

The function may have a point of inflection

The function is decreasing

The function is constant

The function is undefined

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 $g'(x) > -1$

$e^{-x}$

No such values exist

$x = \frac{n\pi}{2}$ where $n$ is an integer

The function is undefined

The function is constant

The function is decreasing

The function may have a local extremum

k(s) results in a vertical tangent line at $s=4$.

k(s) has a critical point at $s=4$.

k(s) has a derivative that does not exist only at the point $s=4$.

k(s) has a horizontal tangent line at $s=4$.