Counting Stationary Points
Count solutions of the derivative equation
“No stationary points” means has no solutions in the domain. If the derivative is quadratic, use its discriminant: negative means no real roots, zero means one repeated root, positive means two distinct roots. These count stationary inputs, not roots of the original curve.
Worked example
For which p does y = x³ + px² + 3x have no stationary points?
Answer
At the derivative has a repeated root: one stationary point remains. These boundary values are excluded.
Your turnPractice
For which real values of does have two distinct stationary points?
Show worked answer
The derivative is . Its discriminant must be positive: . Hence or .