Let $\mathbb F$ be an ordered field, and let $a,b,q,r\in{\mathbb F}$ with $a \lt b$. Let $f(x)=x^3+qx+r$.
Is it true that if $f(a)>0,f(b)>0$, and the discriminant $\Delta=-4p^3 - 27q^2$ of $f$ is negative, then $f(x)>0$ for any $x\in [a,b]$ ?
This property is true whenever $\mathbb F$ is a subfield of $\mathbb R$, because then we have IVT and the sign of $f'$ gives us the variations of $f$.
This holds in any ordered field. To prove it, you can pass to the real closure $K$ of $\mathbb{F}$. Since the discriminant is negative, $f(x)$ has at most one root in any ordered field extending $\mathbb{F}$, and in particular has only one root in $K$. Moreover, $f(x)\to\pm\infty$ as $x\to\pm\infty$. Real-closed fields satisfy the intermediate value theorem for polynomials, and so it follows that $f(x)$ must be positive for every $c\in K$ between $a$ and $b$. (Alternatively, to use a sledgehammer, real-closed fields are elementarily equivalent to $\mathbb{R}$ and your question is first-order, so since it holds for $\mathbb{R}$ it holds for any real-closed field.)