Consider the function:
`f(x)={{:((alphacosx)/(pi-2x),"if "xne(pi)/(2)),(3,"if "x=(pi)/(2)):}`
Which is continuous at `x=(pi)/(2)`, where `alpha` is a constant.
What is the value of `alpha`?

Options

A. 6
B. 3
C. 2
D. 1

Answer

To find the value of \( \alpha \) such that the function

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

is continuous at \( x = \frac{\pi}{2} \), we need to ensure that the limit of \( f(x) \) as \( x \) approaches \( \frac{\pi}{2} \) equals \( f\left(\frac{\pi}{2}\right) \).

### Step 1: Evaluate \( f\left(\frac{\pi}{2}\right) \)

Since \( f\left(\frac{\pi}{2}\right) = 3 \), we have:

\[ f\left(\frac{\pi}{2}\right) = 3 \]

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

We need to compute:

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

Substituting \( x = \frac{\pi}{2} \):

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

This gives us the indeterminate form \( \frac{0}{0} \).

### Step 3: Apply L'Hôpital's Rule

Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that:

\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]

where \( f(x) = \alpha \cos x \) and \( g(x) = \pi - 2x \).

#### Differentiate the numerator and denominator:

- The derivative of the numerator \( f(x) = \alpha \cos x \) is \( f'(x) = -\alpha \sin x \).
- The derivative of the denominator \( g(x) = \pi - 2x \) is \( g'(x) = -2 \).

### Step 4: Compute the limit using L'Hôpital's Rule

Now we compute the limit:

\[ \lim_{x \to \frac{\pi}{2}} \frac{-\alpha \sin x}{-2} = \frac{\alpha \sin\left(\frac{\pi}{2}\right)}{2} \]

Since \( \sin\left(\frac{\pi}{2}\right) = 1 \):

\[ \lim_{x \to \frac{\pi}{2}} f(x) = \frac{\alpha}{2} \]

### Step 5: Set the limit equal to \( f\left(\frac{\pi}{2}\right) \)

For continuity at \( x = \frac{\pi}{2} \):

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

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

Multiplying both sides by 2:

\[ \alpha = 6 \]

### Conclusion

Thus, the value of \( \alpha \) is:

\[ \boxed{6} \]