The cubic has exactly two distinct real roots for some positive value of . What is that value of ?

Answer and explanation

Correct answer: E.

  1. Step 1: Let . Then , with a local maximum at and a local minimum at .
  2. Step 2: and .
  3. Step 3: The equation has exactly two distinct real roots when equals a turning value: or , so or . The positive value is .
Answer
.
Why not the others: would make the horizontal line pass between the turning values ( and ), giving three roots; and also give three roots (since lies between and ); gives one root ().

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