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We are given the function \( f(x) = 2x^2 + 3 \), and we are tasked with finding the absolute maximum and minimum values on two intervals: 
1. \( [2, 7] \)
2. \( [-7, 7] \).

### Step 1: Finding the critical points
We first find the derivative of the function to locate any critical points:

\[
f'(x) = \frac{d}{dx}(2x^2 + 3) = 4x
\]

Set the derivative equal to zero to find the critical points:
\[
4x = 0 \quad \Rightarrow \quad x = 0
\]

### Step 2: Evaluate the function at critical points and endpoints
Next, we evaluate the function \( f(x) \) at the critical point \( x = 0 \) and at the endpoints of each interval.

#### Interval \( [2, 7] \)
- At \( x = 2 \):
  \[
  f(2) = 2(2)^2 + 3 = 2(4) + 3 = 8 + 3 = 11
  \]
- At \( x = 7 \):
  \[
  f(7) = 2(7)^2 + 3 = 2(49) + 3 = 98 + 3 = 101
  \]
- At the critical point \( x = 0 \):
  \[
  f(0) = 2(0)^2 + 3 = 0 + 3 = 3
  \]

For the interval \( [2, 7] \), the **absolute maximum value** is \( 101 \) at \( x = 7 \), and the **absolute minimum value** is \( 3 \) at \( x = 0 \).

#### Interval \( [-7, 7] \)
- At \( x = -7 \):
  \[
  f(-7) = 2(-7)^2 + 3 = 2(49) + 3 = 98 + 3 = 101
  \]
- At \( x = 7 \) (already calculated):
  \[
  f(7) = 101
  \]
- At the critical point \( x = 0 \) (already calculated):
  \[
  f(0) = 3
  \]

For the interval \( [-7, 7] \), the **absolute maximum value** is \( 101 \) at both \( x = -7 \) and \( x = 7 \), and the **absolute minimum value** is \( 3 \) at \( x = 0 \).

### Final Answers:

#### Part (A) \( [2, 7] \):
- Absolute maximum value: \( 101 \) at \( x = 7 \)
- Absolute minimum value: \( 3 \) at \( x = 0 \)

#### Part (B) \( [-7, 7] \):
- Absolute maximum value: \( 101 \) at \( x = -7 \) and \( x = 7 \)
- Absolute minimum value: \( 3 \) at \( x = 0 \)

Let me know if you'd like more details or further clarification.

### Related Questions:
1. How do you find critical points for more complex functions?
2. What role does the derivative play in determining maximum and minimum values?
3. Can there be more than one critical point in a quadratic function?
4. How do absolute maximum and minimum differ from local extrema?
5. How would the analysis change for a cubic function?

### Tip:
Always check both critical points and endpoints when looking for absolute extrema on a closed interval.

Find the absolute maximum and minimum values of the function f(x) = 2x^2 + 3 over the intervals [2, 7] and [-7, 7].

Learn how to find the absolute maximum and minimum values of the quadratic function f(x) = 2x^2 + 3 on the intervals [2, 7] and [-7, 7]. This detailed solution includes derivative calculations, critical points, and evaluations at endpoints.

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