Answer
See explanations.
Work Step by Step
We will need to translate each statement into simplified terms and compare their meanings.
i). Given $f(x)=x^3-3x-1$, to find the roots, we need to solve the equation $x^3-3x-1=0$
ii). Given $y=x^3$ and $y=3x+1$, they will intersect when $x^3=3x+1$ and the x-coordinates are the solutions of the equation $x^3-3x-1=0$
iii) Given $y=x^3-3x$, the function will cross the line $y=1$ when $x^3-3x=1$ and the x-coordinates of the intersection are the solutions of the equation $x^3-3x-1=0$
iv) Given $g(x)=x^4/4-3x^2/2-x+5$, we have $g'(x)=x^3-3x-1$. Letting $g'(x)=0$, we have $x^3-3x-1=0$ and the x-values satisfying this equation are the same as the solutions of the equation $x^3-3x-1=0$
Thus, we conclude that the above statements all ask for the solutions of the equation $x^3-3x-1=0$ .
