5.5 Using the Candidates Test to Determine Absolute (Global) Extrema

Learning objectives

—FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.5• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives.• FUN-4.A.3 Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.• Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.

For a function on a closed interval $[a,b]$ , an absolute extremum can occur only at an interior critical point or at an endpoint. If the function is continuous on $[a,b]$ , the Extreme Value Theorem also guarantees that an absolute minimum and maximum exist. The Candidates Test locates them by comparing all possible outputs.

Find every $c$ in $(a,b)$ where $f'(c)=0$ or $f'(c)$ does not exist, provided $f(c)$ exists.
Add the endpoints $a$ and $b$ to the candidate list.
Evaluate the original function $f$ at every candidate.
The greatest output is the absolute maximum value; the least is the absolute minimum value. State each value and the input where it occurs.

For $f(x)=x^2+2x-3$ on $[-3,2]$ , $f'(x)=2x+2=0$ gives the interior critical point $x=-1$ . The candidates are therefore $-3$ , $-1$ , and $2$ .

Candidate $x$	$f(x)$	Conclusion
$-3$	$0$	neither global extreme
$-1$	$-4$	absolute minimum
$2$	$5$	absolute maximum

Compare values of $f$ , not values of $f'$ . Endpoints must be checked even though the usual interior derivative test does not classify them as local critical points. If the largest or smallest output occurs at several candidates, report every location.

Next Topic

—5.6 Determining Concavity of Functions over Their DomainsRead