Let f(x) be a differentiable function in the interval (0, 2) then the value of `int_(0)^(2)f(x)dx`

Options

A. f(c )for some `cin(0,2)`
B. 2f(c )for some `cin(0,2)`
C. f'(c )for some `cin(0,1)`
D. none of these

Answer

B

Explanation

Let us consider a function `g(t)=int_(0)^(t)f(x)dx`
Now applying Lagrange’s Mean value theorem in (0,2)
`rArr(g(2)-g(0))/(2-0)=g'(c )`, where `cin(0,2)rArrint_(0)^(2)f(x)dx=2f(c ),` where `cin(0,2)`.