The range of value's of k for which the equation ` 2 cos^(4) x - sin^(4) x + k = 0 ` has atleast one solution is ` [ lambda, mu]`. Find the value of ` ( 9 mu + lambda) `.

Answer

To solve the problem, we need to find the range of values of \( k \) for which the equation

\[ 2 \cos^4 x - \sin^4 x + k = 0 \]

has at least one solution. We will then find \( 9\mu + \lambda \) where \( [\lambda, \mu] \) is the range of \( k \).

### Step 1: Rewrite the equation
We start by rewriting the equation:

\[ 2 \cos^4 x - \sin^4 x + k = 0 \]

Using the identity \( \sin^2 x + \cos^2 x = 1 \), we can express \( \sin^4 x \) in terms of \( \cos^4 x \):

\[ \sin^4 x = (1 - \cos^2 x)^2 = 1 - 2\cos^2 x + \cos^4 x \]

Substituting this into the equation gives:

\[ 2 \cos^4 x - (1 - 2 \cos^2 x + \cos^4 x) + k = 0 \]

### Step 2: Simplify the equation
Now, simplify the equation:

\[ 2 \cos^4 x - 1 + 2 \cos^2 x - \cos^4 x + k = 0 \]

This simplifies to:

\[ \cos^4 x + 2 \cos^2 x - 1 + k = 0 \]

Rearranging gives:

\[ \cos^4 x + 2 \cos^2 x + (k - 1) = 0 \]

### Step 3: Let \( y = \cos^2 x \)
Let \( y = \cos^2 x \). The equation now becomes:

\[ y^2 + 2y + (k - 1) = 0 \]

This is a quadratic equation in \( y \).

### Step 4: Determine the discriminant
For the quadratic equation to have at least one real solution, the discriminant must be non-negative:

\[ D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (k - 1) \geq 0 \]

Calculating the discriminant:

\[ 4 - 4(k - 1) \geq 0 \]

This simplifies to:

\[ 4 - 4k + 4 \geq 0 \implies 8 - 4k \geq 0 \implies 4k \leq 8 \implies k \leq 2 \]

### Step 5: Find the minimum value of \( k \)
Next, we need to find the minimum value of \( k \). The quadratic \( y^2 + 2y + (k - 1) \) must also have a valid range for \( y \), which is \( [0, 1] \) since \( y = \cos^2 x \).

For \( y = 0 \):

\[ 0^2 + 2(0) + (k - 1) = k - 1 \geq 0 \implies k \geq 1 \]

For \( y = 1 \):

\[ 1^2 + 2(1) + (k - 1) = 1 + 2 + (k - 1) \geq 0 \implies k + 2 \geq 0 \implies k \geq -2 \]

### Step 6: Combine the results
From the above, we have:

\[ -2 \leq k \leq 2 \]

Thus, the range of \( k \) is:

\[ [\lambda, \mu] = [-2, 2] \]

### Step 7: Calculate \( 9\mu + \lambda \)
Now, we need to calculate \( 9\mu + \lambda \):

\[ \mu = 2, \quad \lambda = -2 \]

Calculating:

\[ 9\mu + \lambda = 9(2) + (-2) = 18 - 2 = 16 \]

### Final Answer
The value of \( 9\mu + \lambda \) is:

\[ \boxed{16} \]