The vaue of `lamda ` so that the function f defined as `f = {{:(lamda x"," , x le pi),(cos x"," ,x gtpi):}` is cotinuous at `x = pi` is :

Options

A. `-1`
B. `- pi`
C. `- (1)/(pi)`
D. `0`

Answer

To find the value of \( \lambda \) such that the function \( f \) defined as

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

is continuous at \( x = \pi \), we need to ensure that the left-hand limit, right-hand limit, and the function's value at that point are all equal.

### Step-by-Step Solution:

1. **Identify the function values at \( x = \pi \)**:
- For \( x < \pi \), \( f(x) = \lambda \).
- For \( x \geq \pi \), \( f(x) = \cos x \).
- Therefore, \( f(\pi) = \cos(\pi) = -1 \).

2. **Calculate the left-hand limit as \( x \) approaches \( \pi \)**:
- The left-hand limit is given by \( \lim_{x \to \pi^-} f(x) = \lambda \).

3. **Calculate the right-hand limit as \( x \) approaches \( \pi \)**:
- The right-hand limit is given by \( \lim_{x \to \pi^+} f(x) = \cos(\pi) = -1 \).

4. **Set the left-hand limit equal to the right-hand limit**:
- For continuity at \( x = \pi \), we need:
\[ \lambda = -1 \]

5. **Conclusion**:
- The value of \( \lambda \) that makes the function continuous at \( x = \pi \) is:
\[ \lambda = -1 \]