the maximum value of `f(x)=x/(4+x+x^2)` on `[-1,1]` is (i) `-1/4` (ii)`-1/3` (iii)`1/6` (iv)`1/b`

Options

A. `-1/3`
B. `-1/4`
C. `1/4`
D. `1/6`

Answer

D

Explanation

Given `f(x)= x/(4+x+x^(2))`
Let `f(x)=1/u` then `u=(4+x+x^(2))/x=4/x+1+x`
`:.(du)/(dx)=-4/(x^(2))+1,(d^(2)u)/(dx^(2))=8/(x^(3))`
For maximum or minimum put `(du)/(dx)=0`
`implies1-4/(x^(2))=0impliesx=+-2`
At `x=-2, (d^(2)u)/(dx^(2))=-8/((2)^(3))lt0`, maxima
At `x=2, (d^(2)u)/(dx^(2))=1gt0`, minima
`:.` At `x=2,f(x)` is maxima
and at `x=-2,f(x)` is minima
`:.` The maximum value at `x=1` is
`f(1)=1/(4+1+1)=1/6`