Answer
All 5 statements ask for the same information, which is to find $x$ so that $x^3-3x-1=0$
Work Step by Step
(a) Find the roots of $f(x)=x^3-3x-1$
Roots of a function $f(x)$ are the values of $x$ such that $f(x)=0$. So basically, the statement asks to find $x$ so that $f(x)=0$.
(b) Find the $x$-coordinates of the points where the curve $y=x^3$ crosses the line $y=3x+1$.
To find the points of intersection, we set up an equation between 2 functions:
$$x^3=3x+1$$ $$x^3-3x-1=0$$
So to find the $x$-coordinates of the points of intersections means to solve this equation, which is the same as to find $x$ so that $f(x)=x^3-3x-1=0$.
(c) Find all the values of $x$ for which $x^3-3x=1$
$$x^3-3x=1$$ $$x^3-3x-1=0$$
Thus, the statement now also comes back to finding $x$ so that $f(x)=x^3-3x-1=0$.
(d) Find the $x$-coordinates of the points where the cubic curve $y=x^3-3x$ crosses the line $y=1$
Again, like (b), finding points of intersection means setting up an equation between 2 functions: $$x^3-3x=1$$ $$x^3-3x-1=0$$
So again, to find the $x$-coordinates of the points of intersection goes back to finding all $x$ for which $f(x)=x^3-3x-1=0$
(e) Solve the equation $x^3-3x-1=0$
Solve the equation is a different way to say finding $x$ so that $f(x)=x^3-3x-1=0$
Therefore, overall, all 5 statements actually ask for the same information.
