A few questions about the graph of $y=x^3-4x+1$.

Question

a) Find an equation for the tangent line to the curve at the point (2,1).

b) What is the range of the values of the curve's slope?

c) Find equations for the tangent lines to the curve at the points where the slope of the tangent line is 8.

I know that asking questions on math.stackexchange that only want answers is not allowed. Therefore, can someone please walk me through some part of this question? Or at least help me start this question? Thank you! Any help is appreciated.

Update: Okay, so I know that I need to find the derivative of this equation in order to get a slope. Using any of the linear equation equations, I plug in (2,1) and come up with the equation of the tangent line.

I do not know what extremal values nor do I know what is the domain of interest.

Finally, I do know what to do for the third hint, but I think I need to figure out b) first in order to do c).

Answer 1

You have $f(x)=x^3-4x+1$ and the derivative $$f'(x)=3x^2-4.$$

As you said yourself in the question, you know that $f'(x)$ is the slope of the tangent line in the point $(x,f(x))$.

• Since $f'(2)=8$, you can see that the tangent line at the point $(2,1)$ is $$y=8x-15.$$
• You want the range of the values of $f'(x)=3x^2-4$. This is quadratic function - and you are probably already familiar with quadratics. You should be able to plot the graph or use $x^2\ge 0$ to see that $$f'(x)=3x^2-4 \ge -4$$ so $f'(x)$ can only attain values from the interval $[-4,\infty)$. And from the properties of quadratic functions you should be able to see that all these values are also attained.
• If the slope is equal to $8$ then you have $$3x^2-4=8$$ which you can solve to get $x=\pm2$ and $f(x)=1$. So this gives you $$y=8x+1\pm16,$$ i.e., $y=8x-15$ and $y=8x+17$. (The first one is precisely the same tangent line as in the first part.)

You can check at least some of the answers in WA:

• tangent line of x^3-4x+1 at x=2
• tangent line of x^3-4x+1 at x=-2
• plot x^3-4x+1, 8x-15, 8x+17

Answer 2

Hints:

1. Use the derivative.

2. Find extremal values of the derivative on the domain of interest.

3. Solve equation $f'(x)=8$ and plug the solutions into the equation of the tangent.