Let `f (x)= {{:((1- tan x)/(4x-pi), x ne (pi)/(4)),( lamda, x =(pi)/(4)):}, x in [0, (pi)/(2)),` If f (x) is continuous in `[0, (pi)/(2))` then `lamda` is equal to:

Options

A. 1
B. `1/2`
C. `-1/2`
D. `-1`

Answer

To determine the value of \(\lambda\) such that the function

\[ f(x) = \begin{cases} \frac{1 - \tan x}{4x - \pi} & \text{if } x \neq \frac{\pi}{4} \\ \lambda & \text{if } x = \frac{\pi}{4} \end{cases} \]

is continuous on the interval \([0, \frac{\pi}{2})\), we need to ensure that the limit of \(f(x)\) as \(x\) approaches \(\frac{\pi}{4}\) equals \(f\left(\frac{\pi}{4}\right) = \lambda\).

### Step 1: Find the limit of \(f(x)\) as \(x\) approaches \(\frac{\pi}{4}\).

We need to calculate:

\[ \lim_{x \to \frac{\pi}{4}} f(x) = \lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{4x - \pi} \]

### Step 2: Substitute \(x = \frac{\pi}{4}\) into the limit.

Substituting \(x = \frac{\pi}{4}\):

- The numerator becomes \(1 - \tan\left(\frac{\pi}{4}\right) = 1 - 1 = 0\).
- The denominator becomes \(4\left(\frac{\pi}{4}\right) - \pi = \pi - \pi = 0\).

Since both the numerator and denominator approach 0, we can apply L'Hôpital's Rule.

### Step 3: Apply L'Hôpital's Rule.

Differentiate the numerator and denominator:

- The derivative of the numerator \(1 - \tan x\) is \(-\sec^2 x\).
- The derivative of the denominator \(4x - \pi\) is \(4\).

Now we can rewrite the limit:

\[ \lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{4x - \pi} = \lim_{x \to \frac{\pi}{4}} \frac{-\sec^2 x}{4} \]

### Step 4: Evaluate the limit.

Substituting \(x = \frac{\pi}{4}\):

\[ \sec^2\left(\frac{\pi}{4}\right) = \left(\frac{1}{\cos\left(\frac{\pi}{4}\right)}\right)^2 = \left(\frac{1}{\frac{\sqrt{2}}{2}}\right)^2 = 2 \]

Thus, we have:

\[ \lim_{x \to \frac{\pi}{4}} \frac{-\sec^2 x}{4} = \frac{-2}{4} = -\frac{1}{2} \]

### Step 5: Set the limit equal to \(\lambda\).

For continuity at \(x = \frac{\pi}{4}\):

\[ \lambda = \lim_{x \to \frac{\pi}{4}} f(x) = -\frac{1}{2} \]

### Conclusion

Thus, the value of \(\lambda\) that makes \(f(x)\) continuous on \([0, \frac{\pi}{2})\) is:

\[ \lambda = -\frac{1}{2} \]