Supremum and infimum of set $A=\{\,x \in\mathbb R : (x - a) (x - b) (x - c) (x - d) < 0\,\}$., where $a < b < c < d$

Question

$\sup\{\,x \in\mathbb R : (x - a) (x - b) (x - c) (x - d) < 0\,\}$, where $a < b < c < d$

Answer 1

Hint: What happens for $x=d$? What happens for all $x>d$? And what happens for $x<d$ that is just barely smaller than $d$ (say, we still have $x>c$)?

Answer 2

A different hint: $(x-a)(x-b)(x-c)(x-d)$ is the standard way to make a polynomial whose roots are exactly $a$, $b$, $c$, and $d$, of degree four and with $1$ as the leading coefficient.

This knowledge (together with $a<b<c<d$) allows you to sketch the function in enough detail that the set you're speaking of is clearly visible on your sketch!

Answer 3

Hint: Sketch the graph of a polynomial with four roots. Note that since its leading coefficient is positive and its degree is even, we know that its end behaviour is positive for small enough $x$ and large enough $x$ (in other words, if we zoom out enough, the polynomial will look roughly like an upward-facing parabola). For what values of $x$ is this polynomial below the $x$-axis? [It will be the union of two open intervals, which should make it easy for you to compute the least upper bound and greatest lower bound.]