https://file.aiquickdraw.com/mathbot/file/1729294484977figo7hbs.jpg

The equation given in the image is:

\[
y = \frac{x^3}{3} - \frac{x^2}{2} - 6x + \frac{5}{3}
\]

We are asked whether the maximum point on the graph is \((-2, 11)\).

### Steps to verify:
1. **Find the first derivative** of the function to locate the critical points.
   
   \[
   \frac{dy}{dx} = x^2 - x - 6
   \]
   
   Setting the derivative equal to zero to find the critical points:

   \[
   x^2 - x - 6 = 0
   \]
   
   Solving this quadratic equation:

   \[
   (x - 3)(x + 2) = 0
   \]
   
   So, the critical points are \(x = 3\) and \(x = -2\).

2. **Find the second derivative** to determine whether these critical points correspond to a maximum or minimum.
   
   The second derivative is:

   \[
   \frac{d^2y}{dx^2} = 2x - 1
   \]
   
   Evaluate the second derivative at \(x = -2\):

   \[
   \frac{d^2y}{dx^2}(-2) = 2(-2) - 1 = -4 - 1 = -5
   \]
   
   Since the second derivative is negative, \(x = -2\) is a **maximum**.

3. **Calculate the y-coordinate** of the maximum point by substituting \(x = -2\) into the original function:

   \[
   y = \frac{(-2)^3}{3} - \frac{(-2)^2}{2} - 6(-2) + \frac{5}{3}
   \]
   Simplifying this:

   \[
   y = \frac{-8}{3} - \frac{4}{2} + 12 + \frac{5}{3}
   \]
   \[
   y = \frac{-8}{3} - 2 + 12 + \frac{5}{3}
   \]
   \[
   y = \frac{-8 + 5}{3} + 10 = \frac{-3}{3} + 10 = -1 + 10 = 9
   \]

Therefore, the maximum point is \((-2, 9)\), **not** \((-2, 11)\). The statement given is **false**.

Would you like further details or have any questions?

---

Here are 5 follow-up questions:
1. What is the significance of the first derivative in identifying critical points?
2. How can you confirm whether a point is a maximum or minimum using the second derivative?
3. Can you explain why solving the quadratic equation gives the critical points?
4. How does the second derivative test help us distinguish between maxima and minima?
5. What would the graph look like if the second derivative were positive at \(x = -2\)?

**Tip**: Always verify critical points by substituting them back into the original function to get the correct y-coordinate!

The maximum point in the graph shown above is (-2,11). True or False?

Verify if the maximum point of the function y = (x^3)/3 - (x^2)/2 - 6x + 5/3 is (-2,11). Learn how to find critical points using derivatives and apply the second derivative test to determine maxima and minima. Suitable for Grades 10-12.

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