Solve a inequality for values of $r$ and $\theta$

Question

Once I used the polar tranforms I obtained the next expression.

$-\cos(\theta)+r^2\cos^2(\theta)\sin(\theta)+b\sin(\theta)-r^3\cos^2(\theta)\sin^2(\theta)$

And I need to prove this inequality is positive for all values of $r$ and $\theta$. We also know that $b>0$.

I've trying to use Mathematica to solve this inequality $-\cos(\theta)+r^2\cos^2(\theta)\sin(\theta)+b\sin(\theta)-r^3\cos^2(\theta)\sin^2(\theta)$ $\forall (r,\theta)$.

Wich kind of command I should use in my computer or what kind of analysis I should use to find this values.

Answer 1

It sounds like you have the inequality $$ -Ar^3 + Br^2 + C \ge 0 $$ where $A,B,C$ are functions of $b,\theta$, that you will need to solve for $r$.

For the special case of $A = 0$, this is quadratic in $r$ and is easily factorable. For $A \neq 0$, divide by $A$ to get $$ r^3 + xr^2 + y \ge 0 $$

Apply the cubic formula to factor this (you can find the detailed instructions here: http://www.sosmath.com/algebra/factor/fac11/fac11.html)