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

Given `F(x)=(int_(a)^(x)f(t)dt-int_(x)^(b)f(t)dt)(2x-(a+b))`………….1
As `f` is continuous `F(x)`is also continuous.
Now, `F(a)=(-int_(a)^(b)f(t)dt)(a-b)=(b-a)int_(a)^(b)f(t)dt`
and `F(b)=(int_(a)^(b)f(t)dt)(b-a)`
Thus `F(a)=F(b)`
Hence, Rolle's Theorem is applicable to `F(x)`.
Therefore there exists at least one `c epsilon(a,b)` such that `F'(c)=0`
`:. 2(int_(a)^(c)f(t)dt-int_(c)^(b)f(t)dt)+(f(c)-(-f(c))).(2c-(a+b))=0`
Hence proved.