Let `f(x) {(|x-2| +a",","if" x le 2),(4x^(2) + 3x +1",","if" x gt 2):}`. If f(x) has a local minimum at x=2, then

Options

A. `a gt 21`
B. `a le 21`
C. `a gt 30`
D. `a gt 24`

Answer

To determine the value of \( a \) such that the function \( f(x) \) has a local minimum at \( x = 2 \), we need to analyze the function defined as follows:

\[ f(x) = \begin{cases} |x - 2| + a & \text{if } x \leq 2 \\ 4x^2 + 3x + 1 & \text{if } x > 2 \end{cases} \]

### Step 1: Calculate \( f(2) \)

For \( x = 2 \), we use the first case of the function since \( 2 \leq 2 \):

\[ f(2) = |2 - 2| + a = 0 + a = a \]

### Step 2: Calculate \( f(2^+) \)

Next, we need to find the value of the function as \( x \) approaches 2 from the right (i.e., \( x > 2 \)). We use the second case of the function:

\[ f(2^+) = 4(2^2) + 3(2) + 1 = 4(4) + 6 + 1 = 16 + 6 + 1 = 23 \]

### Step 3: Set up the condition for local minimum

For \( f(x) \) to have a local minimum at \( x = 2 \), we need:

\[ f(2) \leq f(2^+) \]

Substituting the values we found:

\[ a \leq 23 \]

### Step 4: Calculate \( f(2^-) \)

Now, we need to check the left-hand limit (i.e., as \( x \) approaches 2 from the left):

\[ f(2^-) = f(2) = a \]

### Step 5: Check the derivative condition

To ensure that \( f(x) \) has a local minimum at \( x = 2 \), we also need to check the derivative condition. The derivative from the left (\( f'(2^-) \)) and from the right (\( f'(2^+) \)) should satisfy:

1. \( f'(2^-) \) should be non-positive.
2. \( f'(2^+) \) should be non-negative.

Calculating the derivatives:

- For \( x \leq 2 \):

\[ f'(x) = \frac{d}{dx}(|x - 2| + a) = \begin{cases} -1 & \text{if } x < 2 \\ 0 & \text{if } x = 2 \end{cases} \]

So, \( f'(2^-) = -1 \).

- For \( x > 2 \):

\[ f'(x) = \frac{d}{dx}(4x^2 + 3x + 1) = 8x + 3 \]

Calculating at \( x = 2 \):

\[ f'(2^+) = 8(2) + 3 = 16 + 3 = 19 \]

### Step 6: Conclusion

Since \( f'(2^-) = -1 \) (which is less than 0) and \( f'(2^+) = 19 \) (which is greater than 0), we confirm that \( f(x) \) has a local minimum at \( x = 2 \).

Thus, the final condition we derived is:

\[ a \leq 23 \]

### Summary of Conditions

To ensure \( f(x) \) has a local minimum at \( x = 2 \):
- \( a \) must satisfy \( a \leq 23 \).