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To solve the problem, we need to find the second derivative \( f''(x) \) of the given function \( f(x) = x^4 + 2x^3 - 2x + 1 \), and then evaluate it at \( x = 3 \).

### Step 1: First derivative \( f'(x) \)
Differentiate the given function with respect to \( x \):
\[
f'(x) = \frac{d}{dx} \left( x^4 + 2x^3 - 2x + 1 \right)
\]
Using basic differentiation rules:
\[
f'(x) = 4x^3 + 6x^2 - 2
\]

### Step 2: Second derivative \( f''(x) \)
Now, differentiate \( f'(x) \) with respect to \( x \):
\[
f''(x) = \frac{d}{dx} \left( 4x^3 + 6x^2 - 2 \right)
\]
\[
f''(x) = 12x^2 + 12x
\]

### Step 3: Evaluate \( f''(3) \)
Substitute \( x = 3 \) into \( f''(x) \):
\[
f''(3) = 12(3)^2 + 12(3)
\]
\[
f''(3) = 12(9) + 12(3)
\]
\[
f''(3) = 108 + 36 = 144
\]

So, the correct answer is \( 144 \).

Let me know if you'd like further clarification or a deeper explanation!

Here are 5 related questions:
1. What is the physical interpretation of the second derivative?
2. How would the answer change if the function had an additional term like \( 5x^2 \)?
3. How does differentiating a polynomial function work for higher-order terms?
4. Can you explain how the power rule works in differentiation?
5. What are common mistakes when calculating second derivatives?

**Tip**: Always simplify your derivative expressions before substituting values to minimize errors!

Given the function f(x) = x^4 + 2x^3 - 2x + 1, calculate f''(3).

Learn how to find the second derivative of the polynomial function f(x) = x^4 + 2x^3 - 2x + 1, and evaluate f''(3). This solution walks you through the differentiation process and simplifies the answer step-by-step. Suitable for Grades 10-12 students studying calculus.

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