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Candidates Test — Definition, Formula & Examples

The Candidates Test is a method for finding the absolute maximum and absolute minimum of a continuous function on a closed interval. You evaluate the function at every critical point inside the interval and at both endpoints, then pick the largest and smallest values.

Let ff be continuous on a closed interval [a,b][a, b]. The absolute extrema of ff on [a,b][a, b] occur either at critical points in (a,b)(a, b) where f′(c)=0f'(c) = 0 or f′(c)f'(c) does not exist, or at the endpoints x=ax = a and x=bx = b. The Candidates Test identifies the absolute maximum as the greatest value and the absolute minimum as the least value among all such candidates.

How It Works

First, find f′(x)f'(x) and solve for all xx in the open interval (a,b)(a, b) where f′(x)=0f'(x) = 0 or f′(x)f'(x) is undefined — these are your critical points. Next, evaluate ff at each critical point and at both endpoints aa and bb. Finally, compare all the resulting function values: the largest is the absolute maximum and the smallest is the absolute minimum on [a,b][a, b]. The Extreme Value Theorem guarantees these exist as long as ff is continuous on the closed interval.

Worked Example

Problem: Find the absolute maximum and absolute minimum of f(x)=x3−3x2+1f(x) = x^3 - 3x^2 + 1 on [−1,4][-1, 4].
Find critical points: Differentiate and set equal to zero.
f′(x)=3x2−6x=3x(x−2)=0  ⟹  x=0 or x=2f'(x) = 3x^2 - 6x = 3x(x - 2) = 0 \implies x = 0 \text{ or } x = 2
Evaluate at candidates: Plug the critical points and both endpoints into ff.
f(−1)=−3,f(0)=1,f(2)=−3,f(4)=17f(-1) = -3,\quad f(0) = 1,\quad f(2) = -3,\quad f(4) = 17
Compare values: Select the largest and smallest from the list.
Answer: The absolute maximum is 1717 at x=4x = 4, and the absolute minimum is −3-3 at both x=−1x = -1 and x=2x = 2.

Why It Matters

The Candidates Test is a standard procedure on the AP Calculus AB and BC exams whenever a problem asks for absolute extrema on a closed interval. Engineers and scientists use the same method to find optimal values — such as maximum stress or minimum cost — within constrained ranges.

Common Mistakes

Mistake: Forgetting to check the endpoints and only reporting critical-point values.
Correction: Absolute extrema on a closed interval can occur at endpoints. Always evaluate ff at x=ax = a and x=bx = b alongside the critical points.

Related Terms

  • Extrema — Absolute and relative max/min values this test finds