Maxima and Minima

Calculus Level 5

Subject to \(x+y=1\) for positive reals, the maximum and local minimum of \[ x^4 y + y^4 x \] can be expressed as \( \frac{a}{b}\) and \( \frac{c}{d}\), respectively, where \(a,b\) and \(c,d\) are pairs of positive coprime integers.

Find \(a+b+c+d\).