Traps in Application of Derivatives

A local maximum is the largest value in a neighbourhood, not necessarily on the entire domain. The global max on [a, b] requires checking endpoints too.

Why: Students find f'(x) = 0 and assume the resulting value is the absolute max or min without comparing with endpoint values.

Critical points include both where f'(x) = 0 and where f'(x) is undefined (cusps, corners, vertical tangents). Missing the latter leads to incomplete analysis.

Why: Students only solve f'(x) = 0 and forget to check points where the derivative fails to exist, such as |x| at x = 0.

The slope of the normal is -1/f'(x1), not 1/f'(x1). Forgetting the negative sign gives a line that is not perpendicular to the tangent.

Why: Students remember that the normal slope involves the reciprocal but forget the negative sign required for perpendicularity.

On a closed interval [a, b], the absolute extremum can occur at an endpoint, not just at interior critical points.

Why: Students focus entirely on solving f'(x) = 0 and forget that the closed interval theorem requires evaluating f at the boundary.

The second derivative test is inconclusive when f''(c) = 0. Students incorrectly conclude it is a point of inflection or apply the test anyway.

Why: Students memorize f''(c) > 0 means min and f''(c) < 0 means max, but do not learn the inconclusive case f''(c) = 0.

Rolle's theorem requires continuity on [a, b], differentiability on (a, b), and f(a) = f(b). Applying it without checking all three conditions leads to invalid conclusions.

Why: Students jump directly to finding c where f'(c) = 0 without verifying that the hypothesis of the theorem is satisfied.