Step by Step
C
Critical points
Critical points occur where f'(x) = 0 or where f'(x) is undefined. These are the candidate locations for local maxima, minima, or points of inflection — but finding them is only the first step, not the final answer.
A
Always check endpoints
When finding absolute (global) extrema on a closed interval [a,b], the maximum or minimum might occur at an endpoint rather than at any critical point in between — always evaluate f(a) and f(b) alongside your critical points and compare all the values.
N
Never forget +C
Every indefinite integral (one without specific bounds) requires a "+C" at the end, representing an unknown constant of integration — since many different functions can share the same derivative, differing only by a constant. Omitting it loses points even when the rest of the integration is correct.
Applied Walkthrough
1
Find the absolute maximum of f(x) = x³ − 3x on the interval [-2, 2]. Start by finding critical points: f'(x) = 3x² − 3 = 0, giving x = 1 and x = -1.
2
Evaluate f at both critical points: f(1) = 1 − 3 = −2, and f(-1) = −1 + 3 = 2.
3
Now — critically — also evaluate f at the endpoints: f(-2) = −8 + 6 = −2, and f(2) = 8 − 6 = 2.
4
Comparing all four values (−2, 2, −2, 2), the absolute maximum is 2, occurring at BOTH x = −1 and x = 2 — a student who forgot to check endpoints would have still found the correct maximum value here by luck, but in many problems the true extremum only shows up at an endpoint.
Exam Application
Exams specifically test these three commonly-missed points: correctly identifying critical points (including where the derivative is undefined, not just zero), remembering to check interval endpoints for absolute extrema, and never omitting +C on indefinite integrals.
⚠ Common Trap
The most common and costly trap is finding critical points, evaluating the function only at those points, and never checking the endpoints — missing the true absolute maximum or minimum when it actually occurs at an endpoint instead.
✓ Quick Self-Check
1. What does the CAN mnemonic stand for?
Critical points, Always check endpoints, Never forget +C.
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2. How do you find critical points?
Set f'(x) = 0 or identify where f'(x) is undefined.
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3. Why must you check endpoints when finding absolute extrema on a closed interval?
Because the true maximum or minimum might occur at an endpoint rather than at an interior critical point.
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4. What must every indefinite integral include?
A "+C" representing the unknown constant of integration.
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5. Why is +C necessary on an indefinite integral?
Because many different functions, differing only by a constant, share the same derivative — so without more information, the constant can't be determined.
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