` if f(x) ={underset( cos x " "x ge 0)((x+lambda)^(2) " " x lt 0).` find possible values of `lambda` such that f(x) has local maxima at x=0

Answer

To find the possible values of \( \lambda \) such that the function \( f(x) \) has a local maximum at \( x = 0 \), we will analyze the function defined as follows:

\[ f(x) = \begin{cases} \cos x & \text{if } x \geq 0 \\ (x + \lambda)^2 & \text{if } x < 0 \end{cases} \]

### Step 1: Find \( f(0) \)
First, we need to evaluate \( f(0) \):
\[ f(0) = \cos(0) = 1 \]

### Step 2: Find the left-hand limit as \( x \) approaches 0 from the left
Next, we will find the left-hand limit of \( f(x) \) as \( x \) approaches 0:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x + \lambda)^2 = (0 + \lambda)^2 = \lambda^2 \]

### Step 3: Find the right-hand limit as \( x \) approaches 0 from the right
Now, we will find the right-hand limit of \( f(x) \) as \( x \) approaches 0:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \cos x = \cos(0) = 1 \]

### Step 4: Ensure continuity at \( x = 0 \)
For \( f(x) \) to have a local maximum at \( x = 0 \), it must be continuous at that point. Therefore, we set the left-hand limit equal to the right-hand limit:
\[ \lambda^2 = 1 \]

### Step 5: Solve for \( \lambda \)
Solving the equation \( \lambda^2 = 1 \) gives us:
\[ \lambda = 1 \quad \text{or} \quad \lambda = -1 \]

### Step 6: Check the condition for local maxima
Next, we need to check if \( f(x) \) has a local maximum at \( x = 0 \) for these values of \( \lambda \). We need to ensure that the left-hand derivative (LHD) is greater than the right-hand derivative (RHD).

1. **Left-hand derivative**:
\[ f'(x) = 2(x + \lambda) \quad \text{(for } x < 0\text{)} \]
Evaluating at \( x = 0 \):
\[ f'(0^-) = 2(0 + \lambda) = 2\lambda \]

2. **Right-hand derivative**:
\[ f'(x) = -\sin x \quad \text{(for } x \geq 0\text{)} \]
Evaluating at \( x = 0 \):
\[ f'(0^+) = -\sin(0) = 0 \]

For \( f(x) \) to have a local maximum at \( x = 0 \):
\[ f'(0^-) > f'(0^+) \implies 2\lambda > 0 \implies \lambda > 0 \]

### Conclusion
Thus, the possible values of \( \lambda \) such that \( f(x) \) has a local maximum at \( x = 0 \) are:
\[ \lambda = 1 \]

### Final Answer
The possible value of \( \lambda \) is \( \lambda = 1 \).

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