The minimum value of `(1-x+x^2)/(1+x+x^2)` for all real values ​​of x is:

Options

A. 0
B. `1//3`
C. 1
D. 3

Answer

B

Explanation

Let `y=(1-x+x^(2))/(1+x+x^(2))=1-(2x)/(1+x+x^(2))`
`=1-(2)/((1)/(x)+1+x)`
`impliesy=1-2/t`
where `t=1/x+1+x`
Now, y is minimum, where `2/t` is max `implies` is min.
`therefore (dt)/(dx)=-(1)/(x^(2))+1=0`
`impliesx=pm1`
`(d^(2)t)/dx^(2)=(2)/(x^(2))gt0,` for `x=1`
`therefore` Minimum value of y is
`1-(2)/(1+1+1)=1-2/3=1/3`