Let be the statement '' and be the statement 'the function has no stationary point'. Which of the following is correct? (Here is a real number.)

Answer and explanation

Correct answer: C. is both a necessary and a sufficient condition for

  1. Step 1: . There is a stationary point exactly when has a real solution, i.e. when .
  2. Step 2: So there is no stationary point exactly when : the statements are equivalent.
Answer
is both a necessary and a sufficient condition for .
Why not the others: For the derivative vanishes at , so fails both statements; no value of separates them.

Practise more Necessary and sufficient conditions on a parameter.

Free questions with an explanation for every one, and full timed TMUA mock tests.