f(x)=(1/2) ∫[0--->x] (x-t)²g(t) dt
=(1/2) ∫[0--->x] (x²-2tx+t²)g(t) dt
=(x²/2) ∫[0--->x] g(t) dt - x∫[0--->x] tg(t) dt +(1/2)∫[0--->x] t²g(t) dt
f '(x)=x∫[0--->x] g(t) dt+(x²/2)g(x)-∫[0--->x] tg(t) dt-x²g(x)+(1/2)x²g(x)
=x∫[0--->x] g(t) dt-∫[0--->x] tg(t) dt
f ''(x)=∫[0--->x] g(t) dt+xg(x)-xg(x)
=∫[0--->x] g(t) dt