Solve the Problem $f ( x ) = 3 x ^ { 3 } - 4 x - 1$

Essay

Solve the problem.
-Explain why the following four statements ask for the same information.
(i) Find the roots of $f ( x ) = 3 x ^ { 3 } - 4 x - 1$
(ii) Find the $x$ -coordinates of the intersections of the curve $y = x ^ { 3 }$ with the line $y = 4 x + 1$ .
(iii) Find the $x$ -coordinates of the points where the curve $y = x ^ { 3 } - 4 x$ crosses the horizontal line $y = 1$ .
(iv) Find the values of $x$ where the derivative of $g ( x ) = \frac { 3 } { 4 } x ^ { 4 } - 2 x ^ { 2 } - x + 5$ equals zero.