Minimizing Resultant Volume

The volume ${\bf V(a,b,c) = \pi * \int_{0}^{1} (x^3 + ax^2 + bx + c)^2 dx = }$ $$

${\bf \pi * \int_{0}^{1} (x^6 + 2a x^5 + (a^2 + 2b)x^4 + 2(ab + c)x^3 + (2ac + b^2)x^2 + 2bcx + c^2) \: dx = }$ $$

${\bf \pi * (\frac{x^7}{7} + \frac{a}{3} x^6 + \frac{a^2 + 2b}{5} x^5 + }$ ${\bf \frac{ab + c}{2} x^4 + \frac{2ac + b^2}{3} x^3 + bc x^2 + c^2 x)|_{0}^{1} = }$

${\bf \pi * (\frac{1}{7} + \frac{a}{3} + \frac{a^2 + 2b}{5} + \frac{ab + c}{2} + \frac{2ac + b^2}{3} + bc + c^2) }$ $$

${\bf \frac{\partial V}{\partial a} = \frac{1}{3} + \frac{2a}{5} + \frac{b}{2} + \frac{2c}{3} = 0 }$ $$

${\bf \frac{\partial V}{\partial b} = \frac{2}{5} + \frac{a}{2} + \frac{2b}{3} + c = 0 }$ $$

${\bf \frac{\partial V}{\partial c} = \frac{1}{2} + \frac{2a}{3} + b + 2c = 0 }$

${\bf \implies }$ $$

${\bf 12a + 15b +20c = -10 }$ $$

${\bf 15a + 20b + 30c = -12 }$ $$

${\bf 4a + 6b + 12 c = -3 }$ $$

${\bf \implies }$ $$

${\bf 3b + 16c = 1 }$ $$

${\bf 10b + 60c = 3 }$ $$

${\bf \implies }$ $$

${\bf c = \frac{-1}{20}, b = \frac{3}{5} \implies a = \frac{-3}{2} }$ $$

and, $$

${\bf \frac{\partial^2 V}{\partial a^2} = \frac{2}{5} > 0 }$ $$

${\bf \frac{\partial^2 V}{\partial b^2} = \frac{2}{3} > 0 }$ $$

${\bf \frac{\partial^2 V}{\partial c^2} = 2 > 0 }$ $$

and, $$

the Hessian matrix ${\bf M = [ \begin{array}{ccc} \frac{2}{5} & 0 & 0 \\ 0 & \frac{2}{3} & 0 \\ 0 & 0 & 2 \\ \end{array} ] }$

${\bf det(M) > 0 \implies }$

we have a minimum value at ${\bf (\frac{-3}{2}, \frac{3}{5}, \frac{-1}{20}) }$ $$

${\bf \implies V = \frac{1}{2800} \pi = \dfrac mn \pi \implies }$ $$

${\bf m + n = 2801. }$