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The Candidates
Test

A systematic method for finding the absolute maximum and absolute minimum of a continuous function on a closed interval.

AP Calculus BC

What Are Absolute Extrema?

An absolute maximum is the largest value a function achieves on an entire interval, and an absolute minimum is the smallest. Unlike local extrema, these are global: no other point on the interval produces a higher (or lower) value.

Absolute Maximum
\(f(c) \geq f(x)\) for all \(x\) in the domain
Absolute Minimum
\(f(c) \leq f(x)\) for all \(x\) in the domain

Why Closed Intervals Matter

The Extreme Value Theorem guarantees that a continuous function on a closed interval \([a, b]\) must attain both an absolute maximum and an absolute minimum somewhere on that interval.

a b Abs Max Abs Min Local Min f(a) f(b)

Where Can Extrema Occur?

On a closed interval \([a, b]\), absolute extrema can only occur at two types of points. These are our "candidates":

Critical Points
Points where \(f'(c) = 0\) or \(f'(c)\) does not exist, and \(c\) is in the open interval \((a, b)\)
Endpoints
The boundary values \(x = a\) and \(x = b\) must always be checked, regardless of the derivative

The Candidates Test: Formal Statement

This method is exhaustive: by the Extreme Value Theorem, one of these candidates must be the absolute max and another must be the absolute min. No further analysis (second derivative test, sign charts) is needed.

Visualizing the Candidates

x y a b
\(f'(x)=0\)
 Abs Max f(a) Abs Min max min

Local vs. Absolute Extrema

Local Extrema Absolute Extrema
Scope Nearby neighborhood Entire interval \([a,b]\)
Location Only at critical points Critical points or endpoints
Guaranteed? Not always Yes, by EVT on \([a,b]\)
Test used 1st/2nd Derivative Test Candidates Test
Endpoints checked? No Always

How to Apply the Test

1
Verify continuity on \([a, b]\). If \(f\) is not continuous, the Candidates Test does not apply.
2
Find \(f'(x)\) and solve \(f'(x) = 0\). Also identify any points where \(f'(x)\) does not exist.
3
Discard any critical numbers outside the open interval \((a, b)\).
4
Evaluate \(f\) at each remaining critical number and at both endpoints \(x = a\) and \(x = b\).
5
Compare values: the largest is the absolute maximum, the smallest is the absolute minimum.

Example 1: \(f(x) = x^3 - 3x + 1\)

Find the absolute extrema of \(f(x) = x^3 - 3x + 1\) on \([-2, 2]\).

1
Derivative: \(f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x-1)(x+1)\)
2
Critical points: \(f'(x) = 0\) when \(x = -1\) and \(x = 1\). Both are in \((-2, 2)\).
3
Evaluate: \(f(-2) = -1\), \(f(-1) = 3\), \(f(1) = -1\), \(f(2) = 3\)

Example 1: Graph

-2 -1 0 1 2 Max: 3 Max: 3 Min: -1 Min: -1

Example 2: \(g(x) = x^{2/3}\)

Find the absolute extrema of \(g(x) = x^{2/3}\) on \([-8, 27]\).

1
Derivative: \(g'(x) = \dfrac{2}{3}x^{-1/3} = \dfrac{2}{3\sqrt[3]{x}}\)
2
Critical point: \(g'(x)\) is undefined at \(x = 0\), and \(g'(x) \neq 0\) for any \(x\). So \(x = 0\) is the only critical number.
3
Evaluate: \(g(-8) = (-8)^{2/3} = 4\), \(g(0) = 0\), \(g(27) = 27^{2/3} = 9\)

Example 3: \(h(x) = \sin(x) + \cos(x)\)

Find the absolute extrema of \(h(x) = \sin(x) + \cos(x)\) on \([0, 2\pi]\).

1
Derivative: \(h'(x) = \cos(x) - \sin(x)\). Set equal to zero: \(\cos(x) = \sin(x)\), so \(\tan(x) = 1\).
2
Critical points in \((0, 2\pi)\): \(x = \dfrac{\pi}{4}\) and \(x = \dfrac{5\pi}{4}\).
3
Evaluate: \(h(0) = 1\), \(h(\pi/4) = \sqrt{2}\), \(h(5\pi/4) = -\sqrt{2}\), \(h(2\pi) = 1\).

Example 3: Graph

0 π/4 π/2 π 5π/4 3π/2 2π
\(\sqrt{2}\)
\(-\sqrt{2}\)
 1 1

Common Mistakes

✕ Forgetting Endpoints
The most frequent error: finding critical points but never evaluating \(f(a)\) and \(f(b)\). The absolute extremum often occurs at an endpoint.
✕ Using on Open Intervals
The Candidates Test requires a closed interval \([a, b]\). On open intervals, absolute extrema may not exist at all.
✕ Missing DNE Points
Critical points include where \(f'(x)\) does not exist (cusps, corners), not just where \(f'(x) = 0\). Don't forget these candidates.

Summary and Comparison

Candidates Test 1st/2nd Deriv. Test
Purpose Find absolute extrema Classify local extrema
Domain Closed interval \([a,b]\) Any differentiable region
Endpoints Always checked Not considered
Requires Continuity on \([a,b]\) Differentiability near \(c\)
Output The largest and smallest \(f\)-values Local max/min classification

Question 1 of 5

What is the first step of the Candidates Test for \(f(x)\) on \([a, b]\)?

Question 2 of 5

Find the absolute maximum of \(f(x) = -x^2 + 4x\) on \([0, 5]\).

Question 3 of 5

The Extreme Value Theorem guarantees absolute extrema exist when \(f\) is:

Question 4 of 5

For \(g(x) = x^3 - 12x\) on \([-3, 4]\), which values of \(x\) are candidates for absolute extrema?

Question 5 of 5

If \(f\) is continuous on \([1, 6]\) and you find \(f(1) = 3\), \(f(3) = 7\), \(f(5) = 2\), and \(f(6) = 5\), what is the absolute minimum?