Have you considered the Discriminant?
Firstly, I believe that -- in each case -- the condition on d is that it can represent any Real number (because symbol d does not change the shape of the graph of function f. Also, it goes away, in the derivative).
If you determine the first derivative of function f, you'll find that it's a quadratic polynomial of the form Ax^2 + Bx + C.
(Note: The symbols A, B, and C represent different coefficients than the given a, b, and c -- they are not the same symbols.)
The coefficients of the quadratic polynomial (A, B, and C) defining f`(x) will themselves be expressed in terms of a, b, and c (the coefficients of function f).
EG:
f(x) = 7ax^3 - 2bx^2 + cx - d
f`(x) = 21ax^2 - 4bx + c
In this quadratic polynomial, the coefficients are expressed in terms of a, b, and c:
A = 21a, B = -4b, and C = c
Going back to your exercise, the horizontal tangents on f(x) occur at the roots of f`(x).
Do you remember the Discriminant of a quadratic polynomial, and the conditions for which A, B, and C produce one, two, or no Real roots?
The Discriminant of Ax^2 + Bx + C is B^2 - 4AC
When B^2 - 4AC is positive, the quadratic polynomial has two distinct Real roots.
When the Discriminant is zero, there is one Real root (repeated).
When the Discriminant is negative, there are no Real roots.
Find the symbolic expressions for A, B, and C -- by calculating f`(x) -- in terms of the given coefficients of f -- that is, in terms of a, b, and c.
You can then state the requested conditions on a, b, and c by writing appropriate inequalities using the Discriminant.
