A cubic function is a polynomial of degree 3; that is it has the form f(x)=ax^3+bx^2+cx+d, where a does not equal to zero.

Question

(a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities.
(b) How many local extreme values can a cubic function have?

Answer 1

If we take the derivative of f(x) and set it equal to zero, we get

 

3ax² + 2bx + c = 0

 

The derivative now becomes a quadratic equation.  If we solved for x from this derivative equation, we would have at most two solutions.  But these solutions will actually be our possible critical points because this is derivative, and not the original quadratic equation.

 

The determine the number of critical values, we look at the discriminant of the derivative:  (b² - 4ac)

 

 

For 2 critical values:

 

b² > 4ac

 

This indicates two distinct roots in the quadratic derivative.

 

 

For one critical values:

 

b² = 4ac

 

This indicates repeating roots in the quadratic derivative.

 

 

For no critical values:

 

b² < 4ac

 

This indicates complex roots in the quadratic derivative.

 

In a cubic function there can only be a maximum of 2 local extreme values.

 

 

Understand?