Some quartic (fourth-degree) polynomials resemble deformed parentheses, in that they only have one stationary point, which is their global maximum/minimum. This makes them share some properties of parentheses, such as the uniqueness of the slope of the tangent over their entire domain. Other quartic polynomials, however, have three stationary points, where one or two of them are global maxima/minima.
How do I predict whether the quartic polynomial will have a single turning point, given its equation only (and not by looking at its graph)?
The quartic polynomial (with real coefficients) $p(x)=x^4+ax^3+bx^2+cx+d$ has a single turning point if its first derivative, $p'(x)=4x^3+3ax^2+2bx+c$, has only one real root (or three identical real roots, in the special case $a=b=c=0$). A cubic polynomial, in turn, has only one real root if its discriminant $\Delta$ is negative$^{(*)}$. For the cubic $Ax^3+Bx^2+Cx+D$, with $A\neq 0$, it has the following expression: $$ \Delta=\frac{4(B^2-3AC)^3-(2B^3-9ABC+27A^2D)^2}{27A^2}. $$ One can, therefore, determine whether a quartic polynomial has only a single turning point by computing the discriminant of its first derivative; the answer will be yes if $\Delta<0$.
$^{(*)}$https://en.wikipedia.org/wiki/Cubic_equation