Question 141
Let $n$ be a non-negative integer, and let $f: [-1, 1] \to \mathbb{R}$ be a twice differentiable function vanishing at the origin. Assuming $f''$ continuous, show that
$$
(2n + 3) \int_{-1}^{1} x^{2n} f(x) \, dx = f''(c), \quad \text{for some } c \text{ in the closed interval } [-1, 1].
$$
$$
(2n + 3) \int_{-1}^{1} x^{2n} f(x) \, dx = f''(c), \quad \text{for some } c \text{ in the closed interval } [-1, 1].
$$