Let `f` be a continuous function on `[a ,b]dot` Prove that there exists a number `x in [a , b]` such that `int_a^xf(t)dt=int_x^bf(t)dtdot`

Explanation

Let `g(x)=int_(a)^(x)f(t)dt-int_(x)^(b)f(t)dt, xepsilon[a,b]`
We have `g(a)=int_(a)^(b)f(t)dt` and `g(b)=int_(a)^(b)f(t)dt`
`:.g(a).g(b)=-[int_(a)^(b)f(t)dt]^(2)le0`
clearly `g(x)` is continuous in `[a,b]` an `g(a).g(b)le0`
It implies that `g(x)` will become zero at least once in `[a,b]`.
Hence `int_(a)^(x)f(t)dt=int_(x)^(b)f(t)dt` for all least one value of `xepsilon[a,b]`