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To solve for the absolute maximum and minimum of the function \( f(x) = 2x^2 - 10x + 7 \) on the interval \( [0, 10] \), we need to follow these steps:

1. **Find the critical points** by taking the derivative of the function and setting it equal to zero.
2. **Evaluate the function** at the critical points and at the endpoints \( x = 0 \) and \( x = 10 \).
3. Compare the values to determine the absolute maximum and minimum.

### Step 1: Find the derivative
The derivative of \( f(x) \) is:

\[
f'(x) = \frac{d}{dx}(2x^2 - 10x + 7) = 4x - 10
\]

Set the derivative equal to zero to find the critical points:

\[
4x - 10 = 0
\]
\[
x = \frac{10}{4} = 2.5
\]

So, the critical point is at \( x = 2.5 \).

### Step 2: Evaluate the function at critical points and endpoints

- At \( x = 0 \):

\[
f(0) = 2(0)^2 - 10(0) + 7 = 7
\]

- At \( x = 2.5 \):

\[
f(2.5) = 2(2.5)^2 - 10(2.5) + 7 = 2(6.25) - 25 + 7 = 12.5 - 25 + 7 = -5.5
\]

- At \( x = 10 \):

\[
f(10) = 2(10)^2 - 10(10) + 7 = 2(100) - 100 + 7 = 200 - 100 + 7 = 107
\]

### Step 3: Determine the absolute maximum and minimum
- The function value at \( x = 0 \) is 7.
- The function value at \( x = 2.5 \) is \(-5.5\).
- The function value at \( x = 10 \) is 107.

Thus:
- The **absolute maximum** occurs at \( x = 10 \), where \( f(10) = 107 \).
- The **absolute minimum** occurs at \( x = 2.5 \), where \( f(2.5) = -5.5 \).

Let me know if you'd like more details or have any questions! 

Here are 5 related questions to extend the concepts:
1. How can you find the local maxima and minima of more complex functions?
2. Why do we evaluate endpoints when searching for absolute extrema on a closed interval?
3. What does the second derivative tell us about concavity at critical points?
4. How would the solution change if the interval were open rather than closed?
5. What happens to the behavior of the function as \( x \to \infty \)?

**Tip:** Always include endpoints when finding absolute extrema on a closed interval, as the extreme values can occur at these points.

Consider the function f(x) = 2x^2 - 10x + 7, 0 <= x <= 10. Find the absolute maximum and minimum of f(x) on this interval.

Learn how to find the absolute maximum and minimum of the quadratic function f(x) = 2x^2 - 10x + 7 on the interval [0, 10]. This problem involves using derivatives to locate critical points and evaluating endpoints. Suitable for students in Grades 10-12.

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