If f(x)=ax2+bx+c,∫04f(x)dx=1,∫04xf(x)dx=2,∫04x2f(x)dx=3,  then in [0,4]  the equation f(x)=0  has

∫04(x2−2x+1)f(x)dx=3−4+1=0 ∫04(x−1)2f(x)dx=0 If f(x)  is positive or negative in [0,4] , the integral can not be zero.∴f(x)  Changes its sign.∴f(x)  has at least one root