Analytical Applications of Differentiation

At x = $\sqrt{10} -2$

At x = $\frac{5}{2}$

At x = $\sqrt{5} +2$

The function has no absolute minimum on [0,5]

The function has no absolute extrema

The function has an absolute maximum at $x = 0$

The function has an absolute minimum at $x = 0$

The function has an absolute minimum at $x = 1$

Apply limits solely for functions with discontinuities that suggest piecewise-defined intervals affecting potential extremas.

Use limits when determining if there are any vertical asymptotes acting as boundaries limiting candidate points within an open interval.

Rely only on limits if there are no critical numbers found through derivative tests across an entire given interval.

Utilize limits exclusively when all possible extremes occur precisely at endpoints rather than anywhere else on an interval.

$f'(c) = 0$ or is undefined.

$f'(c) < 0$.

$f'(c) > 0$.

The value of $f'(c)$ can be any real number.

5

10

9

1

Incorrectly assume that it always gives direct indications of absolute maximum or minimum without comparing values.

Presume zero derivatives sufficient for calling a point an absolute extremum without evaluation.

Think that Candidates' test relies exclusively on the derivative whereas second-order conditions are necessary as well.

Believe that first derivative only needs to check positivity or negativity for this test.

The presence of a local minimum in the interval.

The absence of a lower bound in $h(x)$'s interval.

The presence of a local maximum in the interval.

The absence of an upper bound in $h(x)$'s interval.

Calculate additional derivatives beyond $f'(x)$ at each critical number before comparing values.

Consider solely the highest values of $f(x)$ for each critical number without evaluating $f(a)$ or $f(b)$.

Compare only $f(a)$ and $f(b)$ ignoring evaluations of $f(x)$ at each critical number.

Evaluate $f(a)$, $f(b)$, and $f(x)$ at each critical number then compare these values.

Limits like $f(x)=\frac{1}{x}$ approaching infinity provide information about end behavior and need not necessarily indicate an extreme value.

Asymptotes represent limitations in defined behavior but may also influence extremum location.

Critical points can still occur and should be considered even over open ranges.

Points where $f'(x)$ doesn't exist could potentially mark locations for extremum.

It corresponds to an absolute minimum

The function is constant

It corresponds to an absolute maximum

The function has no extrema