For the​ graph, answer parts​ (a) through​ (e) below.
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Part 1
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f(x)



A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical f(x)-axis labeled from negative 6 to 10 in increments of 1. A ray falls from right to left, starting at (3,4) and passes through (1,0). A smooth curve begins at (3,4), falls from left to right at a decreasing rate to a minimum at (4,3), and then rises at an increasing rate, passing through (6,7).
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Part 1
​(a) Determine the​ x-values where f prime left parenthesis x right parenthesisequals0. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The​ x-value(s) where f prime left parenthesis x right parenthesisequals0 ​is/are xequals
  
enter your response here.
​(Type an integer or a decimal. Use a comma to separate answers as​ needed.)
B.
There are no​ x-values where f prime left parenthesis x right parenthesisequals0.

To determine where \( f'(x) = 0 \), we need to identify the points where the slope of the function \( f(x) \) is zero. These are points where the graph of \( f(x) \) has horizontal tangents, meaning it either reaches a local maximum or minimum.

From the graph description:
- The function has a minimum at \( (4, 3) \).

At this point, the graph is flat (the tangent line is horizontal), so the derivative \( f'(x) = 0 \) at \( x = 4 \).

Thus, the correct answer is:

**A. The x-value where \( f'(x) = 0 \) is \( x = 4 \).**

Would you like further clarification or details on any part? 

Here are 5 additional questions related to this concept:
1. What does the derivative \( f'(x) \) represent graphically?
2. How do local maxima and minima relate to the derivative of a function?
3. What is the second derivative test, and how does it apply to points where \( f'(x) = 0 \)?
4. Can a point where \( f'(x) = 0 \) also be an inflection point?
5. How would you calculate \( f'(x) \) from a given function algebraically?

**Tip:** The points where \( f'(x) = 0 \) often indicate key features like maxima, minima, or critical points.

For the graph, answer parts (a) through (e) below.
A coordinate system has a horizontal x-axis labeled from negative 8 to 8 in increments of 1 and a vertical f(x)-axis labeled from negative 6 to 10 in increments of 1. A ray falls from right to left, starting at (3,4) and passes through (1,0). A smooth curve begins at (3,4), falls from left to right at a decreasing rate to a minimum at (4,3), and then rises at an increasing rate, passing through (6,7). (a) Determine the x-values where f'(x) = 0.

In this problem, you will analyze a graph and determine the x-values where the derivative f'(x) equals 0. The solution involves identifying critical points, where the slope of the graph is zero, using calculus concepts. Suitable for students in Grades 10-12.

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