Expert: Sherry Wallin - 11/30/2008

Question
Consider the third-degree polynomial f(x)=ax^3+bx^2+cx+d, a≠0.  Determine conditions for a, b, c, and d if the graph of f has (a) no horizontal tangents, (b) exactly one horizontal tangent, and (c) exactly two horizontal tangents.  Give an example for each case.

Answer
Bill~
    To obtain a horizontal tangent you need to take the first derivative and find the zeros(roots or solutions) of the resulting polynomial. The horizontal tangent is where the slope is zero. Taking the derivative I get 3ax^2+2bx+c = 0 and then using the quadratic formula I get x = [-b+-sqrt(b^2-4ac)]/3a. Now use the discriminant to determine when the equation does not have a real solution so there are no horizontal tangents and only one horizontal tangent would be when there are two identical real solutions and two horizontal tangents would be when you have 2 unique real solutions. In other words, when b^2-4ac> 0 you will have two solutions and therefore two tangents and when b^2-4ac = 0 you have two identical solutions so you have only one tangent and when b^2-4ac < 0 you have no real solutions only two complex solutions so there will be no horizontal tangents. One way to think about all of this is to realize when the slope is zero the graph has reached either it's local maximum or minimum and it changing slope from either positive slope to negative slope in the case where you have a maximum or the other case where the slope is zero and has changed from negative to positive meaning you nave reached a local minimum.
   Notice when you take the derivative d is a constant so becomes zero. When b^2 = 4ac then you have two identical solutions and when
b^2 > 4ac you have two real and unique (different solutions). When
b^2 < 4ac you have no real solutions and therefore no local max or local min.

I hope this has been a learning situation for you. If it is still unclear send additional questions for me to answer.

Math Prof