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Using the Candidates Test to Determine Absolute (Global) Extrema AB & BC

On a closed interval $[a, b]$ where $f$ is continuous, the absolute extrema are found by evaluating $f$ at every critical point inside $(a, b)$ and at both endpoints, then taking the largest and smallest results. The list is complete because an interior absolute extremum has to be a relative one and therefore critical, leaving only the two ends. For $f(x) = x^{3} - 3x^{2} - 9x + 5$ on $[-2, 6]$ the candidates give $3$, $10$, $-22$, $59$, so the absolute maximum is $59$ at the endpoint $x = 6$ and the absolute minimum is $-22$ at $x = 3$.

Two failures recur. The endpoints get left off, and the largest relative maximum is reported instead: on this very function $x = -1$ is the only relative maximum, with value $10$, and it loses to an endpoint by $49$. And the wrong quantity gets reported, either the location where the extreme value occurs instead of the value itself, or a relative extremum where an absolute one was asked for. The comparison in step 3 is always among values of $f$; values of $f'$ answer a different question, and at interior critical points they are mostly zeros.

f(x) = x³ − 3x² − 9x + 5 ON THE CLOSED INTERVAL [−2, 6]. REL MAX 10 REL MIN −22 THE ENDPOINT WINS CANDIDATE f(x) x = −2 3 x = −1 10 x = 3 −22 MIN x = 6 59 MAX EVERY CRITICAL POINT IN THE OPEN INTERVAL, PLUS BOTH ENDPOINTS. EVALUATE f AT EACH, THEN PICK THE LARGEST AND SMALLEST VALUES. THE LARGEST RELATIVE MAXIMUM IS 10 AND THE ABSOLUTE MAXIMUM IS 59.
Drawn to scale at 45 px per unit across and 2 px per unit up. The green relative maximum is a real feature of the function and it does not win.
SAME FUNCTION, TWO INTERVALS. THE ANSWER BELONGS TO THE INTERVAL. REL MAX ON [−2, 2] ON [−2, 6] ABS MAX 10 AT x = −1 ABS MAX 59 AT x = 6 ABS MIN −17 AT x = 2 ABS MIN −22 AT x = 3 ON THE LEFT AN INTERIOR RELATIVE MAXIMUM WINS. ON THE RIGHT THE SAME POINT IS STILL A RELATIVE MAXIMUM AND THE ABSOLUTE MAXIMUM HAS MOVED TO AN ENDPOINT.
Nothing about the point at $x = -1$ changed between the two panels. Only the list of candidates it had to beat changed.

The work

3 ways in · any order
Lesson
Using the Candidates Test to Determine Absolute (Global) Extrema ›

Runs the Candidates Test as a table of inputs and outputs, explains why interior critical points plus both endpoints is a complete list, shows the comparison being made among values of the function rather than of its derivative, and moves the same function to a second interval to show the absolute maximum change hands from a relative maximum to an endpoint.

Skill check · 10 scenarios
Open lesson →
Diagnostic
10-item topic check ›

Ten items spanning the two failure modes of this topic: searching only among interior critical points and reporting the largest relative maximum, and confusing relative with absolute or reporting the location of an extreme value where the value itself was requested.

Not yet available · 10 items
Coming soon
Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.

Take the diagnostic to identify your misconceptions
Diagnostic first