Question 1195392: Sketch the graph of a single function that has all of the properties listed.
a. Continuous and differentiable for all real numbers
b. f'(x)<0 on (-∞,-4) and(0,5)
c. f'(x)>0 on (-4,0) and (5,∞)
d. f''(x)>0 on (-∞,-1) and (2,∞)
e. f''(x)<0 on (-1,2)
f. f'(-4)=f'(5)=0
g.f''(x)=0 at (-1,11) and (2,10)
any help will be so appreciative this is the only question left on my homework that is stumping me
thank you so much in advance!
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! 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.
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