Suppose f is a function such that `f'(x)=4x^3 and f''(x)=12 x^2`. Which of the following is true ?

Options

A. f has a local maximum at x=0 by the `1^(st)` derivative test
B. f has a local minimum at x=0 by the `1^(st)` derivative test
C. f has a local maximum at x=0 by the `2^(nd)` derivative test
D. f has a local minimum at x=0 by the `2^(nd)` derivative test

Answer

To solve the problem, we need to analyze the given derivatives of the function \( f \). We are provided with:

1. \( f'(x) = 4x^3 \)
2. \( f''(x) = 12x^2 \)

We need to determine whether \( f \) has a local maximum or minimum at \( x = 0 \) and which test (first or second derivative test) confirms our conclusion.

### Step 1: Find Critical Points
To find the critical points, we set the first derivative \( f'(x) \) to zero:

\[ f'(x) = 4x^3 = 0 \]

This equation holds true when:

\[ x^3 = 0 \implies x = 0 \]

Thus, \( x = 0 \) is a critical point.

### Step 2: Analyze the First Derivative
Next, we analyze the sign of \( f'(x) \) around the critical point \( x = 0 \):

- For \( x > 0 \):
\[ f'(x) = 4x^3 > 0 \quad (\text{function is increasing}) \]

- For \( x < 0 \):
\[ f'(x) = 4x^3 < 0 \quad (\text{function is decreasing}) \]

Since \( f'(x) \) changes from negative to positive at \( x = 0 \), this indicates that \( x = 0 \) is a local minimum.

### Step 3: Analyze the Second Derivative
Now, we will use the second derivative test to confirm our findings. We evaluate \( f''(x) \) at the critical point \( x = 0 \):

\[ f''(x) = 12x^2 \]

Calculating at \( x = 0 \):

\[ f''(0) = 12(0)^2 = 0 \]

Since \( f''(0) = 0 \), the second derivative test is inconclusive. However, we can analyze the sign of \( f''(x) \):

- For \( x > 0 \):
\[ f''(x) = 12x^2 > 0 \quad (\text{function is concave up}) \]

- For \( x < 0 \):
\[ f''(x) = 12x^2 > 0 \quad (\text{function is still concave up}) \]

Since \( f''(x) > 0 \) for all \( x \neq 0 \), the function is concave up everywhere, confirming that \( x = 0 \) is indeed a local minimum.

### Conclusion
From our analysis, we conclude that:

- \( f \) has a local minimum at \( x = 0 \).
- This conclusion is confirmed by the first derivative test.

Thus, the correct option is:

**Option 4: \( f \) has a local minimum at \( x = 0 \) by the second derivative test.**