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The given problem asks to solve the inequality \( f'(x) < 0 \), where the function \( f(x) = 2x - x^2 \).

### Step-by-Step Solution

1. **Find the derivative of \( f(x) \):**

   Given:
   \[
   f(x) = 2x - x^2
   \]

   The derivative of \( f(x) \) with respect to \( x \) is:
   \[
   f'(x) = \frac{d}{dx}(2x) - \frac{d}{dx}(x^2) = 2 - 2x
   \]

2. **Set up the inequality \( f'(x) < 0 \):**

   \[
   2 - 2x < 0
   \]

   Simplify the inequality:

   \[
   2 < 2x \implies 1 < x
   \]

   So, \( x > 1 \).

3. **Conclusion:**

   The solution to the inequality \( f'(x) < 0 \) is:

   \[
   x > 1
   \]

This means that the function \( f(x) = 2x - x^2 \) is decreasing when \( x > 1 \).

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### Related Questions:

1. How to find the intervals where a function is increasing or decreasing?
2. What is the significance of the derivative in determining the behavior of a function?
3. How do you find critical points of a function?
4. Can a function have multiple intervals where it is increasing or decreasing?
5. How can you apply the first derivative test to determine local extrema?

**Tip:** Remember, a function is decreasing in intervals where its derivative is negative.

Solve the inequality f'(x) < 0 where f(x) = 2x - x^2.

Learn how to solve the inequality f'(x) < 0 for the function f(x) = 2x - x^2. The solution includes finding the derivative, setting up the inequality, and determining where the function is decreasing. This is suitable for students in Grades 10-12.

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