Let $f:R\to R$ be a function such that it satisfy the functional equation $f(x+y)=f\left(x\right)+f\left(y\right)+xy(x+y)$. If $f\text{'}\left(0\right)=-1$, then which of the following option is correct

A function $f\left(x\right)$ is given by $f\left(x\right)=\frac{5^x}{5^x+5}$, then the sum of the series $f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\dots +f\left(\frac{39}{20}\right)$ is equal to:

Let the function $f:\left[0,1\right]\to \mathrm{ℝ}$ be defined by $f\left(x\right)=\frac{4^x}{4^x+2}$. Then the value of $f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\cdots +f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$  is_________

Let $R$ be the set of all real numbers and let $f$ be a function $R$ to $R$ such that $f\left(x\right)+\left(x+\frac{1}{2}\right)f\left(1-x\right)=1$, for all $x\in R$. Then $2f\left(0\right)+3f\left(1\right)$ is equal to.