Question

Solve the following inequalities
$f^{\prime }\left(x\right)⩽g^{\prime }\left(x\right)iff\left(x\right)=2/x,g\left(x\right)=x-x^3$

Answer 1

$f^{\prime }\left(x\right)\le g^{\prime }\left(x\right)\Rightarrow -\frac{2}{x^2}\le 1-3x^2$

$2\ge 3x^4-x^2\Rightarrow 3x^4-x^2-2\ge 0$

$(3x^2+2)(x^2-1)\ge 0$

$3x^2+2$ is always greater than zero.

$x^2-1\ge 0\Rightarrow x\le -1,x\ge 1$

Hence the solution set of given inequality is $R-(-1,1)$