For the​ graph, answer parts​ (a) through​ (e) below.

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Part 1

-8-6-4-22468-6-4-2246810xf(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 (3,3) and passes through (1,2). A smooth curve begins at (3,3), falls from left to right at a decreasing rate to a minimum at (4,2), and then rises at an increasing rate, passing through (6,6).

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Part 1

​(a) Determine the​ x-values where 

f prime left parenthesis x right parenthesisf′(x)equals=0. 

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 parenthesisf′(x)equals=0 

​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 parenthesisf′(x)equals=0.

To solve part (a), we need to find where the derivative of the function \( f(x) \), denoted \( f'(x) \), equals 0. This occurs at the points where the slope of the function is horizontal (i.e., where the tangent line is flat). These points correspond to the critical points of the function, which are either maximums, minimums, or inflection points.

Looking at the description of the graph:

- A ray starts at \( (3, 3) \) and passes through \( (1, 2) \).
- A smooth curve begins at \( (3, 3) \), falls to a minimum at \( (4, 2) \), and then rises again, passing through \( (6, 6) \).

Since the derivative is 0 at the minimum point of a curve (because the slope of the tangent line at this point is horizontal), the minimum occurs at \( x = 4 \).

Thus, the answer is:
- The x-value where \( f'(x) = 0 \) is \( x = 4 \).

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

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Would you like further details, or do you have any questions?

Here are 5 related questions you might find helpful:
1. What does the derivative \( f'(x) \) tell us about the shape of a graph?
2. How do we identify critical points of a function?
3. What is the significance of a minimum or maximum in a function’s graph?
4. How do we interpret the derivative in terms of increasing or decreasing behavior?
5. What happens to \( f'(x) \) at inflection points?

**Tip**: Always look for points where the graph has horizontal tangents to find where \( f'(x) = 0 \).

For the graph, answer parts (a) through (e) below. Part 1: 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 (3,3) and passes through (1,2). A smooth curve begins at (3,3), falls from left to right at a decreasing rate to a minimum at (4,2), and then rises at an increasing rate, passing through (6,6). Part (a): Determine the x-values where f'(x)=0.

Learn how to determine the x-values where the derivative f'(x) equals 0 using a graph. This problem demonstrates key calculus concepts like critical points, graphing functions, and the first derivative test. Suitable for Grades 11-12 students studying calculus.

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