 Solution

The probability density function is zero outside $[0,1]$. Inside that interval, it rises from $(0,0)$, reaches its only interior maximum at $x=2/3$, and falls to $(1,0)$. The derivative ispositive for $0<x<2/3$ and negative for $2/3<x<1$. Its maximum height is $f(2/3)=16/9$. The second panel of the figure in part (b) shows the function graph on the requested full range, with this mode marked.