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?

$$ f(x) = 7x^4 + kx^3 + 8 $$ has a stationary point $(\alpha,f(\alpha))$ whenever for some real $v,$ $$ g(x) = 7x^4 + kx^3 + 8 + v $$ has a repeated zero $\alpha,$ i.e., $$g(x)=7(x-\alpha)^2(x^2+\beta x+\gamma)\\=7x^4+(7\beta-14\alpha)x^3+(7\alpha-14\alpha\beta+7\alpha^2)x^2+(-14\alpha\gamma+7\alpha^2\beta)x+7\alpha^2\gamma.\\$$
Comparing coefficients gives $$\alpha=0$$ or $$\gamma=\frac12\alpha\beta\\\beta=\frac23\alpha\\k=7(\frac23\alpha)-14\alpha\\\alpha=-\frac{3}{28}k.\\$$
Now, $f(x)=x^3(7x+k)+8,\ $ so $f(x)\rightarrow\infty$ as $x\rightarrow\pm\infty.$
Therefore,

*

*if $k=0,$ $f(x)$ has one stationary point, the minimum point $(0,8),$ so $$f(x)\in[8,\infty);$$

*otherwise, $f(x)$ has two stationary points, the inflection point $(0,8)$ and the minimum point $(-\frac{3}{28}k,8-\frac{27}{87808}k^4),$ so $$f(x)\in[8-\frac{27}{87808}k^4,\infty).$$
