"What value of $b$ and $c$ for the curve $y = x^3 + bx^2 + cx$ give the tangent equation $y = 3x - 4$ in the point $(1, -1)$?"
This is what I do:
The derivative of the function is $3x^2 + 2bx + c$
In the point where $x = 1$, it becomes $3 + 2b + c$. Since the tangent equation is $3x - 4$, $3 + 2b + c = 3$. This means that $2b = -c$.
The answer in the book says that $c = -4$, and $b = 2$. However, shouldn't any value that conforms to $2b = -c$ work? I mean, they all give the slope of 3.