F(x)=(2x-4)/(x^2-4x+8),omg=[3,6],find locol and globol minimum point

To find the local and global minimum points of the function \( F(x) = \frac{2x - 4}{x^2 - 4x + 8} \) over the interval \( [3, 6] \), we will follow these steps:

### 1. Take the derivative of the function \( F(x) \) to find critical points.
The quotient rule for derivatives will be used here, where if \( F(x) = \frac{f(x)}{g(x)} \), then:

\[
F'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\]

For our function, let:
- \( f(x) = 2x - 4 \)
- \( g(x) = x^2 - 4x + 8 \)

Now, we compute the derivatives:
- \( f'(x) = 2 \)
- \( g'(x) = 2x - 4 \)

Applying the quotient rule:

\[
F'(x) = \frac{(2)(x^2 - 4x + 8) - (2x - 4)(2x - 4)}{(x^2 - 4x + 8)^2}
\]

### 2. Simplify \( F'(x) \) and set it equal to zero.
To find critical points, we need to set \( F'(x) = 0 \), which corresponds to the numerator being zero. This equation will provide the critical points.

### 3. Evaluate the function at the critical points and the boundaries of the interval.
Once we have the critical points, we will evaluate \( F(x) \) at those points as well as at the boundaries \( x = 3 \) and \( x = 6 \).

### 4. Compare the values to find the local and global minimums.
Finally, by comparing the values of \( F(x) \) at the critical points and the boundaries, we will determine the local and global minimum points.

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Here are some related questions you might find useful:
1. How do you apply the quotient rule in derivatives?
2. What are critical points and how are they related to extrema?
3. How do you find global extrema on a closed interval?
4. What is the importance of boundary points in optimization?
5. How do you simplify complex rational derivatives?

**Tip**: Always evaluate a function at critical points and boundaries when searching for extrema in a closed interval.

F(x) = (2x - 4) / (x^2 - 4x + 8), omg = [3, 6], find local and global minimum point

Learn how to find the local and global minimum points of the function F(x) = (2x - 4) / (x^2 - 4x + 8) on the interval [3, 6]. This solution involves using the quotient rule, finding critical points through the derivative, and evaluating the function at critical points and boundaries. Suitable for college-level calculus students.

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