Let f(x) [-3x+4 0≤x≤10 10 < x ≤ 20 What is the global minimum of f on [0, 20]

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To find the global minimum of the function \( f(x) \) on the interval \([0, 20]\), we need to evaluate the function at the critical points and the endpoints within the given domain. 

Given:
- \( f(x) = -3x + 4 \) for \( 0 \leq x \leq 10 \)
- \( f(x) = 10 \) for \( 10 < x \leq 20 \)

### Steps to find the global minimum:
1. **Evaluate the function at the endpoints:**
   - For \( x = 0 \): \( f(0) = -3(0) + 4 = 4 \)
   - For \( x = 10 \): \( f(10) = -3(10) + 4 = -30 + 4 = -26 \)
   - For \( x = 20 \): \( f(20) = 10 \) (since the function is constant for \( 10 < x \leq 20 \))

2. **Check for any critical points within the intervals:**
   - For the interval \( 0 \leq x \leq 10 \), \( f(x) = -3x + 4 \) is a linear function with a negative slope, so it decreases as \( x \) increases. Thus, the minimum in this interval is at \( x = 10 \), which we already calculated as \( f(10) = -26 \).
   - For the interval \( 10 < x \leq 20 \), \( f(x) = 10 \) is constant, so there is no minimum within this interval other than the constant value 10.

### Conclusion:
The global minimum occurs at \( x = 10 \) with \( f(10) = -26 \).

**The global minimum of \( f(x) \) on \([0, 20]\) is \(-26\).**

Would you like more details or have any questions?

Here are some related questions:
1. How do you find critical points for a function?
2. What is the significance of the endpoints in finding a global minimum?
3. What does it mean for a function to be piecewise defined?
4. How can you determine if a function is decreasing or increasing on an interval?
5. What are the necessary conditions for a point to be a global minimum?
6. How would the global minimum change if the function's slope were positive?
7. How does the continuity of a function affect finding its global minimum?
8. What role do closed intervals play in optimization problems?

**Tip:** Always consider the endpoints of the domain when determining global extrema on a closed interval.

Explore how to find the global minimum of a piecewise linear function f(x) with step-by-step calculations and analysis. This problem involves evaluating endpoints and critical points within the domain [0, 20]. Suitable for advanced high school students (Grades 11-12).

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