Assertion and reason

Exercise and reference answers

Questions

1. Reason

   A function can reach its largest value at an endpoint without f ′ vanishing there.

   Verdict

   Both are true; the reason explains the assertion.

   Assertion

   Endpoints must be checked when finding an absolute maximum.

2. Reason

   If f ′ keeps its sign through c, then c is a point of inflexion.

   Verdict

   Both are true; the reason explains the assertion.

   Assertion

   Not every critical point is a local extremum.

3. Reason

   f ″(c) < 0 indicates a local maximum.

   Verdict

   The assertion is false and the reason is true.

   Assertion

   The second derivative test always decides the question.

4. Reason

   Your syllabus's definition of a critical point covers that case explicitly.

   Verdict

   Both are true; the reason explains the assertion.

   Assertion

   A point where f is not differentiable can be critical.

Explanation