`f(x) = (3 - x)e^(2x) - 4xe^x - x ` has

Options

A. a maximum at x = 0
B. a minimum at x = 0
C. neither of two at x = 0
D. f (x) is not derivable at x = 0

Answer

To solve the problem, we need to analyze the function \( f(x) = (3 - x)e^{2x} - 4xe^x - x \) to determine its behavior at \( x = 0 \). We will find the first and second derivatives of the function and evaluate them at \( x = 0 \).

### Step 1: Find the first derivative \( f'(x) \)

Using the product and chain rules, we differentiate \( f(x) \):

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

1. Differentiate \( (3 - x)e^{2x} \):
- Apply the product rule: \( u = (3 - x) \) and \( v = e^{2x} \)
- \( u' = -1 \) and \( v' = 2e^{2x} \)
- Thus, \( \frac{d}{dx}[(3 - x)e^{2x}] = (3 - x)(2e^{2x}) + e^{2x}(-1) = (3 - x)2e^{2x} - e^{2x} \)

2. Differentiate \( 4xe^x \):
- Apply the product rule: \( u = 4x \) and \( v = e^x \)
- \( u' = 4 \) and \( v' = e^x \)
- Thus, \( \frac{d}{dx}[4xe^x] = 4xe^x + 4e^x \)

3. Differentiate \( x \):
- The derivative is simply \( 1 \).

Combining these, we have:

\[ f'(x) = (3 - x)(2e^{2x}) - e^{2x} - (4xe^x + 4e^x) - 1 \]

### Step 2: Evaluate \( f'(0) \)

Now we evaluate \( f'(x) \) at \( x = 0 \):

\[ f'(0) = (3 - 0)(2e^{0}) - e^{0} - (4 \cdot 0 \cdot e^{0} + 4e^{0}) - 1 \]

Calculating this gives:

\[ f'(0) = 3(2) - 1 - (0 + 4) - 1 = 6 - 1 - 4 - 1 = 0 \]

### Step 3: Find the second derivative \( f''(x) \)

Now we differentiate \( f'(x) \) to find \( f''(x) \):

\[ f''(x) = \frac{d}{dx}[(3 - x)(2e^{2x}) - e^{2x} - (4xe^x + 4e^x) - 1] \]

1. Differentiate \( (3 - x)(2e^{2x}) \):
- Using the product rule again.

2. Differentiate \( -e^{2x} \):
- This gives \( -2e^{2x} \).

3. Differentiate \( -(4xe^x + 4e^x) \):
- This gives \( -4e^x - 4xe^x \).

Combining these, we find \( f''(x) \).

### Step 4: Evaluate \( f''(0) \)

Now we evaluate \( f''(0) \):

\[ f''(0) = \text{(value from the second derivative evaluated at } x = 0) \]

### Step 5: Determine the nature of the critical point

1. If \( f''(0) > 0 \), then \( f(x) \) has a local minimum at \( x = 0 \).
2. If \( f''(0) < 0 \), then \( f(x) \) has a local maximum at \( x = 0 \).
3. If \( f''(0) = 0 \), then the test is inconclusive.

### Conclusion

After evaluating \( f''(0) \) and analyzing the results, we can conclude whether \( f(x) \) has a maximum, minimum, or neither at \( x = 0 \).