The interval of `a` for which the equation `t a n^2x-(a-4)tanx+4-2a=0` has at least one solution `AAx in [0,pi//4]`
a) `a in (2,3)`
b. `a in [2,3]`
c. `a in (1,4)`
d. `a in [1,4]`

Options

A. ` a in (2,3)`
B. `a in [2, 3]`
C. ` a in (1, 4)`
D. ` a in [1 ,4]`

Explanation

To solve the problem, we need to analyze the quadratic equation given in the question:

\[ \tan^2 x - (a - 4) \tan x + (4 - 2a) = 0 \]

This is a quadratic equation in terms of \(\tan x\). We want to find the values of \(a\) such that this equation has at least one solution for \(x\) in the interval \([0, \frac{\pi}{4}]\).

### Step 1: Identify the conditions for real roots
For the quadratic equation \(Ax^2 + Bx + C = 0\) to have real roots, the discriminant must be non-negative:

\[ D = B^2 - 4AC \geq 0 \]

In our case:
- \(A = 1\)
- \(B = -(a - 4)\)
- \(C = 4 - 2a\)

So, the discriminant \(D\) becomes:

\[ D = (-(a - 4))^2 - 4 \cdot 1 \cdot (4 - 2a) \]

### Step 2: Simplify the discriminant
Calculating the discriminant:

\[ D = (a - 4)^2 - 4(4 - 2a) \]

Expanding this:

\[ D = (a^2 - 8a + 16) - (16 - 8a) = a^2 - 8a + 16 - 16 + 8a = a^2 \]

### Step 3: Set the discriminant condition
Since \(D = a^2\), for the quadratic to have real roots, we need:

\[ a^2 \geq 0 \]

This condition is always satisfied for all real \(a\).

### Step 4: Find the range of \(\tan x\)
Next, we need to find the range of \(\tan x\) for \(x \in [0, \frac{\pi}{4}]\):

\[ \tan x \in [0, 1] \]

### Step 5: Set the roots within the range
The roots of the quadratic equation are given by:

\[ \tan x = \frac{(a - 4) \pm \sqrt{D}}{2} = \frac{(a - 4) \pm a}{2} \]

Calculating the roots:

1. \(\tan x_1 = \frac{(a - 4) + a}{2} = \frac{2a - 4}{2} = a - 2\)
2. \(\tan x_2 = \frac{(a - 4) - a}{2} = \frac{-4}{2} = -2\)

### Step 6: Determine the valid range for \(a\)
For the roots to be valid in the interval \([0, 1]\):

1. For \(a - 2\) to be in \([0, 1]\):
- \(0 \leq a - 2 \leq 1\)
- This gives \(2 \leq a \leq 3\).

2. The root \(-2\) is not valid since it is outside the range \([0, 1]\).

### Conclusion
Thus, the only valid interval for \(a\) is:

\[ a \in [2, 3] \]

### Final Answer
The correct option is **b) \(a \in [2, 3]\)**.