设 $$ex a,b>0,quad 0leq fin calR[a,b],quad int_a^b xf(x) d x=0. eex$$ 试证: $$ex int_a^b x^2f(x) d xleq ab int_a^b f(x) d x; eex$$ 并给出使得下列不等式成立的 (您认为的) 最优数: $$ex int_a^b x^3f(x) d xleq (quad) int_a^b f(x) d x. eex$$
解答: 由 $$ex 0geq int_a^b (x-a)(x-b)f(x) d x=int_a^b x^2f(x) d x-abint_a^b f(x) d x eex$$ 即知 $$ex int_a^b x^2f(x) d xleq ab int_a^b f(x) d x. eex$$ 另外, 我们也有 $$eex ea &quad quad 0geq int_a^b (x-b)^3f(x) d x =int_a^b x^3f(x) d x -3bint_a^b x^2f(x) d x -b^3int_a^b f(x) d x\ & a int_a^b x^3f(x) d x leq 3bcdot abint_a^b f(x) d x+b^3int_a^b f(x) d x,\ &quad quad 0geq int_a^b (x-a)^2(x-b)f(x) d x =int_a^b x^3f(x) d x -(2a+b)int_a^b x^2f(x) d x -a^2bint_a^b f(x) d x\ & a int_a^b x^3f(x) d x leq (2a+b)cdot abint_a^b f(x) d x +a^2bint_a^b f(x) d x. eea eeex$$ 于是 $$ex int_a^b x^3f(x) d x leq minsed{(3a+b)b^2,(3a+b)ab}int_a^b f(x) d x =(3a+b)abint_a^b f(x) d x. eex$$