Calculus 1 Is my shown work correct? Find an equation of the line that is tangent to the graph of f and parallel to the given line. I'll post the equations and my work so far done in the comments

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Calculus 1 Is my shown work correct? Find an equation of the line that is tangent to the graph of f and parallel to the given line. I'll post the equations and my work so far done in the comments

Mathematics
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Find an equation of the line that is tangent to the graph of f and parallel to the given line. \[f(x) =x^{3}+2 \] and \[3x - y + 1 = 0\] This is my solution. Is it correct? \[f^{'}= 3x^2\] given line y =3x + 1 m=3 To find where the 2 slopes are equal set f’ prime = m \[3x^{2}= 3\] both sides divided by 3 \[x^{2}=1\] square root of both sides \[x^{2}=\pm1\] Since 1 is a value for x and y = f(x) To determine y, 1 can be plugged into f(x) = \[(1)^{3}+2 = 5\] So we have the coordinates (1,5) as well as m = 3. I used \[y_{2}-y _{1}= m(x _{2}-x _{1})\] y - 5 = 3(x -1) y - 5 = 3x -3 y = 3x +4
y = 3x+4 is one of the solutions there is another tangent line
your work looks good so far though

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