Three of these are wrong. Find the one that holds.

Exercise and reference answers

Questions

1. Question

   Three of these statements about critical points are false.

   Choices

   1. Item

      f ′(c) = 0 or f is not differentiable at c

      Correct

      Yes

   2. Explanation

      It may be a point of inflexion instead.

      Item

      Every critical point is a local extremum

   3. Explanation

      The DERIVATIVE vanishes, not f.

      Item

      A critical point needs f(c) = 0

   4. Explanation

      They occur wherever f ′ vanishes or fails to exist.

      Item

      Critical points only occur at endpoints

2. Question

   Three of these statements about the second derivative test are false.

   Choices

   1. Item

      It fails when f ′(c) = 0 and f ″(c) = 0

      Correct

      Yes

   2. Explanation

      The test reads the sign of f ″(c), so f ″ must exist.

      Item

      It works without f ″ existing at c

   3. Explanation

      Negative second derivative means a local MAXIMUM.

      Item

      f ″(c) < 0 indicates a local minimum

   4. Explanation

      It fails outright when f ″(c) = 0.

      Item

      It never fails

Explanation