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To solve problem 69, let’s analyze the function \( f(x) = x^4 + x \) and its derivatives. We’ll go through each part of the question to confirm whether \( f \) has a relative extremum or inflection point at \( x = 0 \).

### Step 1: First Derivative \( f'(x) \)

1. Calculate \( f'(x) \):
   \[
   f'(x) = 4x^3 + 1
   \]
   
2. Determine if \( f'(x) = 0 \) at \( x = 0 \):
   \[
   f'(0) = 4(0)^3 + 1 = 1 \neq 0
   \]
   
Since \( f'(0) \neq 0 \), there is no critical point at \( x = 0 \), which means there’s no relative extremum at \( x = 0 \).

### Step 2: Second Derivative \( f''(x) \)

1. Calculate \( f''(x) \):
   \[
   f''(x) = 12x^2
   \]
   
2. Evaluate \( f''(0) \):
   \[
   f''(0) = 12(0)^2 = 0
   \]
   
Although \( f''(0) = 0 \), this alone doesn’t indicate an inflection point. For an inflection point, the concavity of \( f(x) \) needs to change around \( x = 0 \).

### Step 3: Third Derivative \( f'''(x) \)

1. Calculate \( f'''(x) \):
   \[
   f'''(x) = 24x
   \]
   
2. Evaluate \( f'''(0) \):
   \[
   f'''(0) = 24 \cdot 0 = 0
   \]
   
The third derivative also vanishes at \( x = 0 \), which provides no information about the concavity change.

### Step 4: Fourth Derivative \( f^{(4)}(x) \)

1. Calculate \( f^{(4)}(x) \):
   \[
   f^{(4)}(x) = 24
   \]
   
Since \( f^{(4)}(x) \) is a constant (24) and positive, this suggests that \( f(x) \) is always concave up around \( x = 0 \). Thus, there is no change in concavity, meaning there’s no inflection point at \( x = 0 \).

### Conclusion

Even though \( f''(0) = 0 \), the graph of \( f(x) = x^4 + x \) does not have a relative extremum or an inflection point at \( x = 0 \).

### Sketch of \( f(x) \)

The function \( f(x) = x^4 + x \) has a steep slope near \( x = 0 \), as \( f'(0) = 1 \). Since \( x^4 \) grows faster than \( x \) for large \( |x| \), the graph resembles a flattened \( x^4 \)-shaped curve around the origin but inclines due to the \( +x \) term.

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**Related Questions:**
1. What conditions define a relative extremum in a function?
2. How does the second derivative test determine the presence of an inflection point?
3. Why do we look at higher derivatives when the second derivative is zero?
4. Can a function have an inflection point where \( f''(x) = 0 \) but higher derivatives are non-zero?
5. How do different powers of \( x \) affect the shape of a polynomial graph?

**Tip:** Checking higher derivatives is essential when the second derivative test is inconclusive, especially for polynomial functions.

Let f(x) = x^4 + x. Show that even though f''(0) = 0, the graph of f has neither a relative extremum nor an inflection point where x = 0. Sketch the graph of f(x).

This problem demonstrates how to analyze the critical points and inflection points of f(x) = x^4 + x. Even though the second derivative at x = 0 is 0, we show that there is neither a relative extremum nor an inflection point at x = 0. Learn the detailed steps, including first, second, and higher derivative tests.

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