How do I find the minimum of a quartic function with a missing coefficient? $$ f(x) = 7x^4 + kx^3 + 8 $$
I have a question asking me to state the range of this function. To do this it seems I have to find the relationship between k and the minimum of the function. I've been told that there is a way to do this with calculus, but this is an algebra class and I haven't learned any calculus yet. I've tried coming up with random factors and then using the remainder theorem to solve for k, but that just gives me an identity.
How do I find the range of this function without knowing k?
Alternatively, is it possible to find k using the given information?

Using calculus min is at $x=\frac{-3k}{28}$
How to get this without calculus is a question.
