For a continuous function defined by different analytic formulas on adjacent…
MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation
Problem
For a continuous function defined by different analytic formulas on adjacent sub-intervals of a closed domain, which of the following best explains why any absolute maximum or minimum on the domain must be sought not only where f′(x)=0 but also at the sub-interval endpoints and the points where the analytic formulas change?
(a) Because at interior points where an extremum can occur we have f′(x)=0, and endpoints are automatically critical, so testing only those suffices.
(b) Because absolute extrema can occur only where the function is not differentiable, namely at the endpoints and formula-change points.
(c) At those locations the derivative is either zero or fails to exist, so they include every possible critical value within the domain.
(d) Because the Extreme Value Theorem does not apply to piecewise-defined functions, so one must inspect the endpoints and join points separately.