If `sin^4theta+cos^4theta+lambda=0` has a real solution then range of `lambda` is

Options

A. `[-1,1]`
B. `[-1,(-1/2)]`
C. `[1/2, 1]`
D. [-1/2,1/2]`

Answer

To solve the problem, we need to analyze the equation \( \sin^4 \theta + \cos^4 \theta + \lambda = 0 \) and determine the range of \( \lambda \) for which this equation has real solutions.

### Step-by-Step Solution:

1. **Rewrite the Equation**:
We start with the equation:
\[ \sin^4 \theta + \cos^4 \theta + \lambda = 0 \]
Rearranging gives:
\[ \sin^4 \theta + \cos^4 \theta = -\lambda \]

2. **Use the Identity**:
We can use the identity:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta \]
Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we have:
\[ \sin^4 \theta + \cos^4 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \]

3. **Express in Terms of \( \sin^2 \theta \)**:
Let \( x = \sin^2 \theta \). Then \( \cos^2 \theta = 1 - x \) and we can express:
\[ \sin^4 \theta + \cos^4 \theta = 1 - 2x(1-x) = 1 - 2x + 2x^2 \]
Therefore, we can rewrite our equation as:
\[ 1 - 2x + 2x^2 = -\lambda \]
or
\[ 2x^2 - 2x + (1 + \lambda) = 0 \]

4. **Discriminant Condition**:
For this quadratic equation in \( x \) to have real solutions, the discriminant must be non-negative:
\[ D = b^2 - 4ac = (-2)^2 - 4 \cdot 2 \cdot (1 + \lambda) \geq 0 \]
Simplifying gives:
\[ 4 - 8(1 + \lambda) \geq 0 \]
\[ 4 - 8 - 8\lambda \geq 0 \]
\[ -8\lambda \geq 4 \]
\[ \lambda \leq -\frac{1}{2} \]

5. **Finding the Maximum Value of \( \lambda \)**:
The maximum value of \( \lambda \) occurs when the discriminant is zero:
\[ D = 0 \implies 4 - 8(1 + \lambda) = 0 \]
Solving gives:
\[ 4 = 8 + 8\lambda \implies 8\lambda = -4 \implies \lambda = -\frac{1}{2} \]

6. **Finding the Minimum Value of \( \lambda \)**:
The minimum value of \( \lambda \) occurs when \( \sin^2 \theta = 0 \) or \( \cos^2 \theta = 0 \):
\[ \sin^4 \theta + \cos^4 \theta = 1 \]
Thus, we have:
\[ 1 + \lambda = 0 \implies \lambda = -1 \]

7. **Final Range of \( \lambda \)**:
Therefore, the range of \( \lambda \) for which the equation has real solutions is:
\[ -1 \leq \lambda \leq -\frac{1}{2} \]

### Conclusion:
The range of \( \lambda \) is:
\[ \boxed{[-1, -\frac{1}{2}]} \]