Question 1195392
Certainly, let's sketch a function that satisfies all the given properties.

**1. Understand the Properties**

* **a. Continuous and differentiable for all real numbers:** This means the graph has no breaks or sharp corners.
* **b. f'(x) < 0 on (-∞,-4) and (0,5):** The function is decreasing on these intervals (slope is negative).
* **c. f'(x) > 0 on (-4,0) and (5,∞):** The function is increasing on these intervals (slope is positive).
* **d. f''(x) > 0 on (-∞,-1) and (2,∞):** The function is concave up (opens upwards) on these intervals.
* **e. f''(x) < 0 on (-1,2):** The function is concave down (opens downwards) on this interval.
* **f. f'(-4) = f'(5) = 0:** There are horizontal tangents (critical points) at x = -4 and x = 5.
* **g. f''(x) = 0 at (-1,11) and (2,10):** There are inflection points at (-1,11) and (2,10) where concavity changes.

**2. Sketch the Graph**

* **Start with the concavity:**
    * Concave up on (-∞, -1) and (2, ∞)
    * Concave down on (-1, 2)

* **Add the critical points:**
    * Horizontal tangents at x = -4 and x = 5

* **Determine increasing/decreasing intervals:**
    * Decreasing on (-∞, -4) and (0, 5)
    * Increasing on (-4, 0) and (5, ∞)

* **Connect the points smoothly:**
    * Ensure the graph is continuous and differentiable everywhere.
    * Make sure the graph reflects the concavity and increasing/decreasing behavior.

**Here's a rough sketch of a possible function:**

* **(Note: This is just one possible representation. There could be variations that still satisfy all the given properties.)**

   * The graph would start by increasing and concave up from (-∞, -4).
   * At x = -4, it would have a horizontal tangent and continue increasing, but now concave down.
   * At x = -1, there's an inflection point (concavity changes).
   * The graph continues increasing and concave down until x = 0.
   * At x = 0, it has another horizontal tangent and starts decreasing.
   * At x = 2, there's another inflection point (concavity changes).
   * The graph continues decreasing and concave up until x = 5.
   * At x = 5, it has a horizontal tangent and starts increasing and remaining concave up towards positive infinity.

**Key Points:**

* The graph should have the general shape described above, reflecting the given properties of the function's derivatives.
* The exact positions and heights of the points are not fixed, as long as they satisfy the given conditions.

I hope this helps! Let me know if you'd like to explore specific aspects or have further questions.