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Select all values of x where the function f has a critical point such that f′(x)=0; given choices: x= -8, -7,-4,-2,0-1,2,4,6,8

To identify the points where \( f'(x) = 0 \), we need to look at the graph for points where the slope of the tangent line to the curve is zero. These occur at the local maxima, minima, or any points where the function levels off horizontally.

From the graph:

1. At \( x = -4 \), there is a local maximum (the curve flattens and changes direction).
2. At \( x = 2 \), there is a local minimum (the curve flattens and changes direction).
3. At \( x = 6 \), there is a local maximum (the curve flattens and changes direction).

Thus, the function \( f(x) \) has critical points where \( f'(x) = 0 \) at:
- \( x = -4 \)
- \( x = 2 \)
- \( x = 6 \)

The correct values of \( x \) from the given choices are: \( x = -4 \), \( x = 2 \), and \( x = 6 \).

Would you like further clarification or details on this? Here are some related questions you might find interesting:

1. What is the significance of a critical point in calculus?
2. How can you determine if a critical point is a maximum, minimum, or saddle point?
3. Can you explain the relationship between \( f(x) \) and \( f'(x) \) for understanding graph behavior?
4. How does concavity relate to the second derivative \( f''(x) \)?
5. What happens to the graph near points where \( f'(x) = 0 \)?

**Tip:** Critical points are important because they provide insight into the local behavior of a function, especially in optimization problems!

Select all values of x where the function f has a critical point such that f′(x)=0; given choices: x= -8, -7, -4, -2, 0, -1, 2, 4, 6, 8.

Learn how to identify critical points on a function graph by finding where the derivative f'(x) equals zero. In this problem, we analyze points on the graph to determine local maxima and minima, critical for calculus students in Grades 10-12.

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