Let `f(x) = sin^6x + cos^6x + k(sin^4 x + cos^4 x)` for some real number `k`. Determine all real numbers `k` for which `f(x)` is constant for all values of `x`.

Options

A. `-3/4`
B. `3/4`
C. `3/2`
D. `-3/2`

Answer

To determine all real numbers \( k \) for which the function

\[ f(x) = \sin^6 x + \cos^6 x + k(\sin^4 x + \cos^4 x) \]

is constant for all values of \( x \), we can follow these steps:

### Step 1: Rewrite the function using identities
We can use the identity \( \sin^2 x + \cos^2 x = 1 \) to simplify the terms.

Recall that:
\[ \sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x)(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x) = 1(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x) \]

### Step 2: Express \( \sin^4 x + \cos^4 x \)
Using the identity for \( \sin^4 x + \cos^4 x \):
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x \]

### Step 3: Substitute these identities into \( f(x) \)
Substituting these identities back into \( f(x) \):
\[ f(x) = \sin^4 x - \sin^2 x \cos^2 x + \cos^4 x + k(1 - 2\sin^2 x \cos^2 x) \]
This simplifies to:
\[ f(x) = (\sin^4 x + \cos^4 x) - \sin^2 x \cos^2 x + k - 2k \sin^2 x \cos^2 x \]

### Step 4: Combine like terms
Now, we combine the terms:
\[ f(x) = 1 - 3\sin^2 x \cos^2 x + k \]

### Step 5: Set the expression to be constant
For \( f(x) \) to be constant for all \( x \), the coefficient of \( \sin^2 x \cos^2 x \) must be zero. Thus, we set:
\[ -3 + k = 0 \]

### Step 6: Solve for \( k \)
Solving the equation gives:
\[ k = 3 \]

### Step 7: Conclusion
Thus, the value of \( k \) for which \( f(x) \) is constant for all \( x \) is:
\[ \boxed{3} \]