We should consider that the organization expands the yearly membership by \[\mathbf{Rs}\text{ }\mathbf{x}.\]
Along these lines, x is the quantity of supporters who end the administrations.
Absolute income, \[\mathbf{R}\left( \mathbf{x} \right)\text{ }=\text{ }\left( \mathbf{500}\text{ }\text{ }\mathbf{x} \right)\text{ }\left( \mathbf{300}\text{ }+\text{ }\mathbf{x} \right)\]
\[=\text{ }\mathbf{150000}\text{ }+\text{ }\mathbf{500x}\text{ }\text{ }\mathbf{300x}\text{ }\text{ }\mathbf{x2}\]
\[=\text{ }-\text{ }\mathbf{x2}\text{ }+\text{ }\mathbf{200x}\text{ }+\text{ }\mathbf{150000}\]
Separating the two sides w.r.t. x, we get R'(x) \[=\text{ }-\text{ }\mathbf{2x}\text{ }+\text{ }\mathbf{200}\]
For LOCAL maxima and nearby minima, R'(x) = 0
\[-\text{ }\mathbf{2x}\text{ }+\text{ }\mathbf{200}\text{ }=\text{ }\mathbf{0}\Rightarrow \mathbf{x}\text{ }=\text{ }\mathbf{100}\]
\[\mathbf{R}”\left( \mathbf{x} \right)\text{ }=\text{ }-\text{ }\mathbf{2}\text{ }<\text{ }\mathbf{0}\text{ }\mathbf{Maxima}\]
Along these lines, R(x) is most extreme at \[\mathbf{x}\text{ }=\text{ }\mathbf{100}\]
Along these lines, to get most extreme benefit, the organization should expand its yearly membership by \[\mathbf{Rs}\text{ }\mathbf{100}.\]