Expert: Paul Klarreich - 3/12/2011

Question
I would greatly appreciate help on 1 more (since you offered:)

Find conditions on a, b, c, and d so that the graph of the polynomial f(x)=axcubed+bxsquared+cx+d has:
1-exactly two horizontal tangents
2-exactly one horizontal tangent
3-no horizontal tangents

Thanks!

Answer
Questioner: Emily
Country: United States
Category: Calculus
Private: No
Subject: polynomial and tangent lines
Question: I would greatly appreciate help on 1 more (since you offered:)

Find conditions on a, b, c, and d so that the graph of the polynomial f(x)=axcubed+bxsquared+cx+d has:
1-exactly two horizontal tangents
2-exactly one horizontal tangent
3-no horizontal tangents

Thanks!
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You will want to look up questions with the subject line 'Graph Sketching'.  My practice, in answering questions, is to edit the subject line for the questioner.  That makes it easy for others to find answers to questions similar to their own.

OK, enough of that stuff.

For  f(x) = px^3 + qx^2 + rx + s,  your typical cubic, you want critical points.  (Them's your hor. tang.'s)

[I changed a,b,c,d.  You will see the reason later.]

So you will find:

f'(x) = 3px^2 + 2qx + r,

and set that to zero.  It is a quadratic, which could have:

TWO DISTINCT REAL ROOTS.
EXACTLY ONE REAL ROOT. (I.e. the two real roots are equal.)
NO REAL ROOTS.

Remember the quadratic formula and the DISCRIMINANT, which is  b^2 - 4ac?

In your high school intermediate algebra text, you will find:

If b^2 - 4ac > 0, the roots are real and distinct.
If b^2 - 4ac = 0, the roots are real and equal.(i.e. one 'double' root.)
If b^2 - 4ac < 0, the roots are imaginary.

Here is what you will do:

Take a,b,c, which are your  3p, 2q, r, and compute b^2 - 4ac

Then you will...

Oops -- my computer just broke.  You will have to finish it yourself.