Calculate the greatest and least values of the function `f(x)=(x^4)/(x^8+2x^6-4x^4+8x^2+16)`

Explanation

`(1)/(f(x)) = (x^(4) + (16)/(x^(4))) + 2 (x^(2) + (4)/(x^(2))) - 4`
Now, `A.M ge G.M`
`implies x^(4) + (16)/(x^(4)) ge 8` and `x^(2) + (4)/(x^(2)) ge 4`
`implies (1)/(f(x)) ge 12 implies f(x) le (1)/(12)`
Again using `A.M ge G.M` we have
`(2x^(6) + 8 x^(2))/(2) ge 4x^(4)`
or `2 x^(6) + 8 x^(2) - 4 x^(4) ge 4 x^(4) ge 0`
or `x^(8) + 2x^(6) - 4x^(4) + 8x^(2) + 16 lt 0`
Also, `x^(4) ge 0`
`implies (x^(4))/(x^(8) + 2x^(6) - 4x^(8) + 8x^(2) + 16) ge 0`
`implies f(x) ge 0`
Hence, the greatest value is 1/12 and the least value is 0