[Math] How to check which intervals a cubic function will increase and in which intervals it will decrease

derivativesfunctions

I was trying to find the intervals in which the cubic function $4x^3 -6x^2 -72x + 30$ would be strictly increasing and strictly decreasing.

I managed to get the fact that at the values {-2,3} the differential of the function is zero. However this divides the function in three intervals, how can i know which intervals the function increases in and which intervals the function decreases in?

Note: I know you could simply plot the function. I was hoping for a more analytical method.

Best Answer

Whether a differentiable function is increasing or decreasing (or stationary) at a point is (essentially) determined by the sign of its derivative. For a cubic polynomial function $$p(x) = a x^3 + b x^2 + c x + d ,$$ the derivative is $$p'(x) = 3 a x^2 + 2 b x + c .$$ Thus, the character of the roots of $p'$, that is, the critical points of $p$, is determined completely by the discriminant $\Delta := ((2 b)^2) - 4 (3a) (c) = 4 b^2 - 12 a c$ of $p'$:

  • If $\Delta > 0$, then $p'$ has two distinct roots, $r_- < r_+$. If $a > 0$ (in particular, if it is monic), then since $\lim_{x \to \pm \infty} p'(x) = +\infty$, we conclude that $p$ is increasing where $x < r_-$ and $x > r_+$, decreasing where $r_- < x < r_+$, and hence has a local maximum at $r_-$ and a local minimum at $r_+$.
  • If $\Delta = 0$, then $p'$ has a double root, $r$. If $a > 0$, we conclude that $p$ is increasing where $x \neq r$ and that $p'(r) = 0$, so that $p$ has an inflection point at $r$, and has no minimum or maximum. (In fact, $p$ is strictly increasing everywhere, as $x < y$ implies $p(x) < p(y)$.)
  • If $\Delta < 0$, then $p'$ has no roots. If $a > 0$, $p$ is increasing everywhere.

(All of these statements are reversed appropriately when $a < 0$.)

Example For our polynomial $p(x) = 4 x^3 - 6 x^2 - 72 x + 30,$ computing gives $\Delta = 3600 > 0$, and we are in the first case.

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