We are tasked with finding the point(s) on the graph of the function $f(x) = x^3 - 12x + 1$ where the function has a horizontal tangent line.

Step 1: Understand the problem

• A horizontal tangent line occurs when the slope of the tangent line is zero.
• The slope of the tangent line is given by the derivative of the function $f(x)$.

Step 2: Find the derivative of $f(x)$

We differentiate $f(x) = x^3 - 12x + 1$ with respect to $x$:

$f'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(12x) + \frac{d}{dx}(1)$

Using basic differentiation rules: $f'(x) = 3x^2 - 12$

Step 3: Set the derivative equal to zero

To find the points where the tangent line is horizontal, we need to set the derivative equal to zero and solve for $x$:

$3x^2 - 12 = 0$

Step 4: Solve for $x$

Simplify and solve the equation:

$3x^2 = 12$ $x^2 = 4$ $x = \pm 2$

So, the function has horizontal tangent lines at $x = 2$ and $x = -2$.

Step 5: Find the corresponding $y$-values

To find the points, we substitute $x = 2$ and $x = -2$ into the original function $f(x) = x^3 - 12x + 1$:

• For $x = 2$: $f(2) = 2^3 - 12(2) + 1 = 8 - 24 + 1 = -15$

• For $x = -2$: $f(-2) = (-2)^3 - 12(-2) + 1 = -8 + 24 + 1 = 17$

Step 6: Conclusion

The points on the graph where the function has horizontal tangent lines are:

• $(2, -15)$
• $(-2, 17)$

Would you like any additional details or clarifications?

Here are 5 related questions to deepen your understanding:

1. What is the geometric meaning of a horizontal tangent line in terms of the graph's shape?
2. How do we determine whether the horizontal tangent line corresponds to a local maximum or minimum?
3. How does the second derivative of a function help classify critical points (such as those with horizontal tangents)?
4. Can a cubic function have more than two points with horizontal tangent lines?
5. How would the function's behavior change if we altered the constants in $f(x) = x^3 - 12x + 1$?

Tip: When solving for critical points, always check the second derivative to classify the nature of the critical points (whether they are maxima, minima, or points of inflection).