What is the range of function f(x)=x(x^2-1)÷x^4-x^2+1

Answer 1

THe bottom does not factor so the domain is all reals

THe top is x^3 - x

The first derivative is:

[ (x^4 -x^2+1)(3x^2 - 1) - (x^3 -x)(4x^3-2x)]/ ( x^4 -x^2+1)^2

Finding the extrema by setting it equal to zero, the polynomial

in the numerator is:

0 = (x^4 -x^2+1)(3x^2 - 1) - (x^3 -x)(4x^3-2x)

= -4x^6 + 3x^5 - 3x^4 -x^4 +4x^4+2x^4 + 3x^2 + x^2 -2x^2 - 1

= -4x^6 + 3x^5 + 2x^4 + 2x^2 - 1

Rational root theorem says the only possible rational roots are {+-1, +-1/2 +-1/4}

NONE of them work, so the irrational solution must be found numerically.

They occur at x= +- 0.618

the max and min are +- 0.5 respectively

graph it at desmos and you will see