Answer
$(1,2)$ and $(-1,0)$
Work Step by Step
Differentiate the give equation.
$\frac{dy}{dx}=\frac{d}{dx}(x^4-2x^2-x)$
$\frac{dy}{dx}=4x^3-4x-1$
Let the two points be
$(a,b)$ and $(p,q)$
Since the slope at these two points are equal, so
$4a^3-4a-1=4p^3-4p-1$
$a(a^2-1)=p(p^2-1)$
Since $a\neq p$
So $a^2-1=p^2-1$
Since $a\neq p$
Therefore. $p=-a$
Substitute the value of p in $4a^3-4a-1=4p^3-4p-1$
$4a^3-4a-1=4(-a)^3-4(-a)-1$
$a=\pm 1$
Substitute 1 for a in the given equation $y=x^4-2x^2-x$
$y=-2$
Substitute -1 for a in the given equation $y=x^4-2x^2-x$
$y=0$
Therefore the two points are $(1,-2)$ and $(-1,0)$
