CIIM 2021 Problem 6

Let $0 \le a < b$ be real numbers. Prove that there is no continuous function $f : [a, b] \to \mathbb{R}$ such that $$\int_a^b f(x)x^{2n} \mathrm dx>0 \text{and} \int_a^b f(x)x^{2n+1} \mathrm dx <0$$ for every integer $n \ge 0$ .