Let `f` be a continuous function on `[a , b]dot` If `F(x)=(int_a^xf(t)dt-int_x^bf(t)dt)(2x-(a+b)),` then prove that there exist some `c in (a , b)` such that `int_a^cf(t)dt-int_c^bf(t)dt=f(c)(a+b-2c)dot`

Explanation

To prove that there exists some \( c \in (a, b) \) such that

\[ \int_a^c f(t) dt - \int_c^b f(t) dt = f(c)(a + b - 2c), \]

we start by analyzing the given function \( F(x) \):

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