For what value of k, function `f(x)={((k cosx)/(pi-2x)",","if "x ne (pi)/(2)),(3",", "if " x =(pi)/(2)):}` is continuous at `x=(pi)/(2)`?

Options

A. 1
B. 3
C. 5
D. 6

Explanation

To determine the value of \( k \) for which the function

\[ f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x} & \text{if } x \neq \frac{\pi}{2} \\ 3 & \text{if } x = \frac{\pi}{2} \end{cases} \]

is continuous at \( x = \frac{\pi}{2} \), we need to ensure that the limit of \( f(x) \) as \( x \) approaches \( \frac{\pi}{2} \) is equal to the value of the function at that point, which is \( 3 \).

### Step 1: Find the limit as \( x \) approaches \( \frac{\pi}{2} \)

We first compute the limit:

\[ \lim_{x \to \frac{\pi}{2}} f(x) = \lim_{x \to \frac{\pi}{2}} \frac{k \cos x}{\pi - 2x} \]

### Step 2: Evaluate the limit

Substituting \( x = \frac{\pi}{2} \) directly into the function gives:

\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \pi - 2\left(\frac{\pi}{2}\right) = 0 \]

This results in the indeterminate form \( \frac{0}{0} \). Therefore, we can apply L'Hôpital's Rule.

### Step 3: Apply L'Hôpital's Rule

Using L'Hôpital's Rule, we differentiate the numerator and denominator:

- The derivative of the numerator \( k \cos x \) is \( -k \sin x \).
- The derivative of the denominator \( \pi - 2x \) is \( -2 \).

Thus, we have:

\[ \lim_{x \to \frac{\pi}{2}} \frac{k \cos x}{\pi - 2x} = \lim_{x \to \frac{\pi}{2}} \frac{-k \sin x}{-2} = \lim_{x \to \frac{\pi}{2}} \frac{k \sin x}{2} \]

### Step 4: Evaluate the limit again

Now substituting \( x = \frac{\pi}{2} \):

\[ \sin\left(\frac{\pi}{2}\right) = 1 \]

So, we have:

\[ \lim_{x \to \frac{\pi}{2}} \frac{k \sin x}{2} = \frac{k \cdot 1}{2} = \frac{k}{2} \]

### Step 5: Set the limit equal to the function value

For the function to be continuous at \( x = \frac{\pi}{2} \), we need:

\[ \frac{k}{2} = 3 \]

### Step 6: Solve for \( k \)

Multiplying both sides by \( 2 \):

\[ k = 6 \]

### Conclusion

The value of \( k \) for which the function \( f(x) \) is continuous at \( x = \frac{\pi}{2} \) is \( k = 6 \).

---