Let `f:[-1,2]->[0,oo)` be a continuous function such that `f(x)=f(1-x)fora l lx in [-1,2]dot` Let `R_1=int_(-1)^2xf(x)dx ,` and `R_2` be the area of the region bounded by `y=f(x),x=-1,x=2,` and the x-axis . Then (a)`R_1=2R_2` (b) `R_1=3R_2` (c)`2R_1` (d) `3R_1=R_2`

Options

A. `R_(1)=2R_(2)`
B. `R_(1)=3R_(2)`
C. `2R_(1)=R_(2)`
D. `3R_(1)=R_(2)`

Explanation

To solve the problem, we need to analyze the given function \( f(x) \) and the integrals \( R_1 \) and \( R_2 \).

### Step 1: Understanding the properties of \( f(x) \)
Given that \( f(x) = f(1 - x) \) for all \( x \) in the interval \([-1, 2]\), this symmetry will be crucial in our calculations.

### Step 2: Set up the integral for \( R_1 \)
The integral \( R_1 \) is defined as:
\[
...