Find the points on the graph of 𝑓(𝑥)=48𝑥−𝑥3f⁡(x)=48⁢x−x3 where the tangent line is horizontal.

(Use symbolic notation and fractions where needed. Give your answer as a comma separated list or ordered pairs in the form (∗,∗), (∗,∗), (∗,∗)…)

To find the points where the tangent line is horizontal, we need to determine where the derivative of the function \( f(x) = 48x - x^3 \) is equal to zero. A horizontal tangent line occurs when the slope of the function (i.e., the derivative) is zero.

### Steps:

1. **Differentiate \( f(x) \):**

   The derivative of \( f(x) = 48x - x^3 \) is:

   \[
   f'(x) = \frac{d}{dx}(48x - x^3) = 48 - 3x^2
   \]

2. **Set the derivative equal to zero:**

   To find the points where the tangent line is horizontal, set \( f'(x) = 0 \):

   \[
   48 - 3x^2 = 0
   \]

3. **Solve for \( x \):**

   \[
   3x^2 = 48
   \]
   \[
   x^2 = 16
   \]
   \[
   x = \pm 4
   \]

4. **Find the corresponding \( y \)-coordinates:**

   To find the \( y \)-coordinates, substitute \( x = 4 \) and \( x = -4 \) into the original function \( f(x) = 48x - x^3 \):

   - For \( x = 4 \):
     \[
     f(4) = 48(4) - (4)^3 = 192 - 64 = 128
     \]

   - For \( x = -4 \):
     \[
     f(-4) = 48(-4) - (-4)^3 = -192 - (-64) = -192 + 64 = -128
     \]

### Final Answer:
The points where the tangent line is horizontal are:
\[
(4, 128), (-4, -128)
\]

Would you like further details or have any questions?

Here are 5 related questions you might explore:
1. What is the behavior of the graph around \( x = 4 \) and \( x = -4 \)?
2. How can we determine whether these points are local maxima or minima?
3. What is the second derivative of the function \( f(x) = 48x - x^3 \)?
4. How does the cubic term \( -x^3 \) affect the shape of the graph?
5. How can you find the inflection points of \( f(x) \)?

**Tip:** The sign of the second derivative can help classify the critical points as maxima or minima.

Find the points on the graph of f(x)=48x−x^3 where the tangent line is horizontal.

Discover how to find the points where the tangent line to the curve of f(x) = 48x − x^3 is horizontal. This calculus problem explores derivatives, critical points, and the application of horizontal tangent lines. Ideal for students in Grades 10-12.

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