Home > Calculus > Identifying Stationary Points (Critical Points) for a Function

How many stationary points can a cubic function have?

Answer 1

Evelyn Agard

A cubic polynomial with real coefficients can have at most 2 real stationary points

We'll restrict our consideration to cubic polynomials with real coefficients in this case.

Examine the general cubic polynomial equation:

#f(x) = Ax^3+Bx^2+Cx+D# #{A,B,C,D}in RR#

Now consider #f'(x)#

#f'(x) = 3Ax^2+2Bx+C#

The stationary points of #f(x)# will be where #f'(x)=0#

#f'(x)# is a quadratic that will have 2 real or complex roots. The real roots may be co-incident.

Hence, #f(x)# will have at most 2 real stationary points.

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Answer 2

Adam Adkins

A cubic function can have at most two stationary points.

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Answer 3

Christopher Agresta

A cubic function can have a maximum of two stationary points. These stationary points occur where the derivative of the cubic function is equal to zero. Since a cubic function is a polynomial of degree three, its derivative is a quadratic function, which can have at most two real roots. Each real root corresponds to a stationary point on the cubic function's graph. Therefore, a cubic function can have up to two stationary points.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.