$($a$)$ Find the maximum value of fsx$1,$ x$2,...,$ xnd$$s nx$1$x$2$xn given that x$1,$ x$2,...,$ xn are positive numbers and x$11$x$211$xn$$c$,$ where c is a constant. $($b$)$ Deduce from part $($a$)$ that if x$1,$ x$2,...,$ xn are positive numbers, then s nx$1$x$2$xn