Examples of Problem Solving

Pl[x1_,x2_]:=Plot[{-Log[x],-Log[x]*x},
{x,x1,x2},PlotStyle->{Blue,Magenta}];
Va[x_]:=Graphics[{Blue,Text[Style[N[-Log[x]*x],20],
{x+.1,-Log[x]+.1}],Opacity[.3],Polygon[{{0,0},
{x,0},{x,-Log[x]},{0,-Log[x]}}]}];
xmax=Solve[D[-Log[x]*x,x]\[Equal]0,x][[1,1,2]];
Vmax=Graphics[{Red,Text[Style[-Log[xmax]*xmax,30],
{xmax+.05,-Log[xmax]+.1}],Text[Style["*",30],
{xmax,-Log[xmax]*xmax-.1}],Opacity[.5],
Polygon[{{0,0},{xmax,0},
{xmax,-Log[xmax]},{0,-Log[xmax]}}]}];
Show[Pl[0,1.1],Vmax,Va[.15],Va[.65],
ImageSize->600,GridLines->Automatic]