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Find the interval of values of x for which the function is decreasing.

To find the interval where the function \( f(x) = 10 - x^2 \) is decreasing, we can follow a similar procedure:

1. **Find the derivative \( f'(x) \)**:
   The first derivative of the function will indicate the rate of change of \( f(x) \). The function is decreasing where the derivative is negative.

   \[
   f'(x) = \frac{d}{dx} \left( 10 - x^2 \right)
   \]

   \[
   f'(x) = -2x
   \]

2. **Determine where \( f'(x) < 0 \)**:
   The function is decreasing where the derivative is less than zero.

   \[
   -2x < 0
   \]

   \[
   x > 0
   \]

Thus, the function \( f(x) = 10 - x^2 \) is decreasing on the interval \( (0, \infty) \).

Would you like any further details or have any questions?

Here are some related questions you might consider:
1. What is the interval where the function is increasing?
2. What are the critical points of the function \( f(x) = 10 - x^2 \)?
3. How do you determine the maximum value of this function?
4. What is the vertex of the parabola described by \( f(x) = 10 - x^2 \)?
5. How can you determine the axis of symmetry of the function \( f(x) = 10 - x^2 \)?

**Tip:** When analyzing quadratic functions like \( f(x) = 10 - x^2 \), remember that they represent parabolas. The sign of the \( x^2 \) term tells you if the parabola opens upwards (minimum point) or downwards (maximum point).

Discover how to find the interval where the function f(x) = 10 - x^2 is decreasing with a detailed step-by-step solution. This problem involves calculus concepts, including derivatives and function analysis. Suitable for students in Grades 11-12.

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