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Existence of constant (5)

Let f:[0, 1] \rightarrow \mathbb{R} be a continuous function such that

(1)   \begin{equation*} \int_0^1 f(x) \, \mathrm{d}x = \int_0^1 x f(x) \,\mathrm{d}x \end{equation*}

Prove that there exists a c \in (0, 1) such that

    \[\int_0^1 xf(x) \, \mathrm{d}x = \int_c^1 f(x) \, \mathrm{d}x\]

Solution

Define F(t) := \int_t^1 f(x)\, \mathrm{d}x. Then, integrating by parts it follows that

    \[\int_0^1 xf(x)\, \mathrm{d}x = \int_0^1 F(x)\, \mathrm{d}x\]

By the mean value theorem for integrals, there is a point c in (0,1) such that \int_0^1 F(x)\, dt = F(c), i.e.

    \[\int_0^1 xf(x)\, \mathrm{d}x = \int_c^1 f(x)\, \mathrm{d}x\]

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