E(Y) = -2*Y**5/25 + 3*Y**4/10 + 2*Y**3/5 - 8*Y**2/5 - 8*Y/5 + 209/50
E(1) = 8/5   E(9/4) = 143/1280 = 0.11171875
number of real roots of E in [1, 9/4] (Sturm): 0
number of real roots of E - 1/100 in [1, 9/4]: 0   (E-1/100)(1) = 159/100
=> E(s) > 1/100 > 0 for all s in [0,1] (since Y = sqrt(1+4s) ranges over [1, sqrt5] and sqrt5 < 9/4).
