Off[General::spell,  General::spell1]
    
v = A x^2; 
    v2 = D[v, {x, 2}]
      
W = Integrate[EI v2^2/2 - w v, {x, 0, L}]
    vl = v/.Solve[
            D[W, A] == 0, A][[1]]

vl2 = D[vl, {x, 2}];
nvlw = vl /. valuew;
nvl2w = vl2 /. valuew;
Plot[{nvlw, -nvl2w}, {x, 0, 1}, 
      PlotStyle -> {GrayLevel[0.6], GrayLevel[0]}]