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 $a+b+c+d$ .