For the function f(x) = 3x3 - 4x- + 11x+8, find f''(x). Then find f'(0) and f''(5).

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f''(x)=

Let's first find the derivatives step by step.

### Given:
\[ f(x) = 3x^3 - 4x^2 + 11x + 8 \]

#### Step 1: Find \( f'(x) \)
To find the first derivative \( f'(x) \), apply the power rule to each term:

\[
f'(x) = \frac{d}{dx}(3x^3) - \frac{d}{dx}(4x^2) + \frac{d}{dx}(11x) + \frac{d}{dx}(8)
\]

\[
f'(x) = 9x^2 - 8x + 11
\]

#### Step 2: Find \( f''(x) \)
Now, take the derivative of \( f'(x) \) to find \( f''(x) \):

\[
f''(x) = \frac{d}{dx}(9x^2) - \frac{d}{dx}(8x) + \frac{d}{dx}(11)
\]

\[
f''(x) = 18x - 8
\]

### Now, let's calculate the values:

#### Step 3: Find \( f'(0) \)
Substitute \( x = 0 \) into \( f'(x) = 9x^2 - 8x + 11 \):

\[
f'(0) = 9(0)^2 - 8(0) + 11 = 11
\]

#### Step 4: Find \( f''(5) \)
Substitute \( x = 5 \) into \( f''(x) = 18x - 8 \):

\[
f''(5) = 18(5) - 8 = 90 - 8 = 82
\]

### Final answers:
- \( f''(x) = 18x - 8 \)
- \( f'(0) = 11 \)
- \( f''(5) = 82 \)

Would you like more details on any of these steps?

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**Relative questions:**
1. What is the significance of the second derivative in a function?
2. How would the critical points be found for \( f(x) \)?
3. How does the first derivative test determine local extrema?
4. What would be the concavity of \( f(x) \) when \( x = 5 \)?
5. How does \( f'(x) \) relate to the graph of \( f(x) \)?

**Tip:** The second derivative, \( f''(x) \), helps determine concavity and inflection points in a graph.

For the function f(x) = 3x^3 - 4x^2 + 11x + 8, find f''(x). Then find f'(0) and f''(5).

Discover how to find the first and second derivatives of the function f(x) = 3x^3 - 4x^2 + 11x + 8, along with calculating f'(0) and f''(5). This calculus problem covers key concepts such as applying the power rule and finding critical values. Suitable for students in Grades 10-12.

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