Question

on a separate sheet of paper, graph f(x)=x³ - 3x + 1 by using a table of values. then, estimate the x - coordinates at which each zero and relative extrema occur, and state the domain and range. zeros between x = select choice and x = select choice, between x = select choice and x = select choice, and between x = select choice and x = select choice relative maximum at x = select choice relative minimum at x = select choice d = select choice, r = select choice need help with this question? get a hint

Explanation:

Step1: Create a table of values

Choose some values of \(x\) such as \(x=- 3,-2,-1,0,1,2,3\) and calculate \(y = f(x)=x^{3}-3x + 1\).
When \(x=-3\), \(y=(-3)^{3}-3\times(-3)+1=-27 + 9+1=-17\).
When \(x=-2\), \(y=(-2)^{3}-3\times(-2)+1=-8 + 6+1=-1\).
When \(x=-1\), \(y=(-1)^{3}-3\times(-1)+1=-1 + 3+1=3\).
When \(x = 0\), \(y=0^{3}-3\times0+1=1\).
When \(x = 1\), \(y=1^{3}-3\times1+1=1 - 3+1=-1\).
When \(x = 2\), \(y=2^{3}-3\times2+1=8 - 6+1=3\).
When \(x = 3\), \(y=3^{3}-3\times3+1=27-9 + 1=19\).

Step2: Find zeros

A zero of the function \(y = f(x)\) is a value of \(x\) for which \(y = 0\). By looking at the sign - change of \(y\) values in the table, the function changes sign between \(x=-2\) and \(x=-1\), between \(x = 0\) and \(x = 1\), and between \(x = 1\) and \(x = 2\).

Step3: Find the derivative

The derivative of \(y=f(x)=x^{3}-3x + 1\) is \(y'=f'(x)=3x^{2}-3\).
Set \(y' = 0\), then \(3x^{2}-3=0\), which simplifies to \(x^{2}-1=0\), so \(x=\pm1\).

Step4: Determine relative extrema

Take the second - derivative \(y''=f''(x)=6x\).
When \(x=-1\), \(y''=6\times(-1)=-6<0\), so there is a relative maximum at \(x=-1\).
When \(x = 1\), \(y''=6\times1=6>0\), so there is a relative minimum at \(x = 1\).

Step5: Determine domain and range

The domain of a polynomial function \(y=x^{3}-3x + 1\) is all real numbers, \(D=(-\infty,\infty)\).
Since the function is a cubic function with a positive leading coefficient (\(a = 1\) in \(y=ax^{3}+bx^{2}+cx + d\)), the range is also all real numbers, \(R=(-\infty,\infty)\).

Answer:

Zeros between \(x=-2\) and \(x=-1\), between \(x = 0\) and \(x = 1\), and between \(x = 1\) and \(x = 2\).
Relative maximum at \(x=-1\).
Relative minimum at \(x = 1\).
\(D=(-\infty,\infty)\), \(R=(-\infty,\infty)\)